or, How to Remember Trigonometric Identities
revised 25 Jan 2012
Copyright © 1997-2012 Stan Brown, Oak Road Systems
Summary: Faced with the large number of trigonometric identities, students tend to try to memorize them all. That way lies disaster. When you memorize a formula by rote, you have no way to know whether you’re remembering it correctly. I believe it is much more effective (and, in the long run, much easier) to understand thoroughly how the trig functions work, memorize half a dozen formulas, and work out the rest as needed. That’s what these pages show you how to do.
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Trig without Tears Part 1:
Trigonometry is fascinating! It started as the measurement (Greek metron) of triangles (Greek trigonon), but now it has been formalized under the influence of algebra and analytic geometry and we talk of trigonometric functions. not just sides and angles of triangles.
Trig is almost the ideal math subject. Big and complex enough to have all sorts of interesting odd corners, it is still small and regular enough to be taught thoroughly in a semester. (You can master the essential points in a week or so.) It has lots of obvious practical uses, some of which are actually taught in the usual trig course. And trig extends plenty of tentacles into other fields like complex numbers, logarithms, and calculus.
If you’d like to learn some of the history of trigonometry and peer into its dark corners, I recommend Trigonometric Delights by Eli Maor (Princeton University Press, 1998).
The computations in trigonometry used to be a big obstacle. But now that we have calculators, that’s no longer an issue.
Would you believe that when I studied trig, back when dinosaurs ruled the earth (actually, in the 1960s), to solve any problem we had to look up function values in long tables in the back of the book, and then multiply or divide those five-place decimals by hand? The “better” books even included tables of logs of the trig functions, so that we could save work by adding and subtracting five-place decimals instead of multiplying and dividing them. My College Outline Series trig book covered all of plane and spherical trigonometry in 188 pages—but then needed an additional 138 pages for the necessary tables!
Though calculators have freed us from tedious computation, there’s still one big stumbling block in the way many trig courses are taught: all those identities. They’re just too much to memorize. (Many students despair of understanding what’s going on, so they just try to memorize everything and hope for the best at exam time.) Is it tan²A + sec²A = 1 or tan²A = sec²A + 1? (Actually, it’s neither—see equation 39!)
Fortunately, you don’t need to memorize them. This paper shows you the few that you do need to memorize, and how you can produce the others as needed. I’ll present some ideas of my own, and a wonderful insight by W.W. Sawyer.
I wrote Trig without Tears to show that you need to memorize very little. Instead, you learn how all the pieces of trigonometry hang together, and you get used to combining identities in different ways so that you can derive most results on the fly in just a couple of steps.
You might like to read some ideas of mine on the pros and cons of memorizing.
To help you find things, I’ll number the most important equations and other facts. (Don’t worry about the gaps in the numbering. I’ve left those to make it easier to add information to these pages.)
A very few of those, which you need to memorize, will be marked “memorize:”. Please don’t memorize the others. The whole point of Trig without Tears is to teach you how to derive them as needed without memorizing them. If you can’t think how to derive one, the boxes should make it easy to find it. But then please work through the explanation. I truly believe that if you once thoroughly understand how all these identities hang together, you’ll never have to memorize them again. (It’s worked for me since I first studied trig in 1965.)
By the way, I love explaining things but sometimes I go on a bit too long. So I’ve put some interesting but nonessential notes in a separate page and inserted hyperlinks to them at appropriate points. If you follow them (and I hope you will), use your browser’s “back” command to return to the main text.
Much as it pains me to say so, if you’re pressed for time you can still get all the essential points by ignoring those side notes. But you’ll miss some of the fun.
Trig without Tears concerns itself exclusively with plane trigonometry, which is what’s taught today in nearly every first course. Spherical trigonometry is not dealt with.
I’m also restricting myself to real arguments to the functions and real values of the functions. I have to draw the line somewhere! (I do use real-valued functions with the polar form of complex numbers in the Notes.) For complex trigonometric functions, see chapter 14 of Eli Maor’s Trigonometric delights (Princeton University Press, 1998).
In Trig without Tears we’ll work with identities and solve triangles. A separate (and much shorter) page of mine explains how to solve trigonometric equations.
Finally, there are a lot of good math sites out there: some are good, some less good. I’ve collected a dozen or so that you may find useful.
I’ve made some compromises since many common math characters can’t be displayed in a standard way:
In talking about the domains and ranges of functions, it is handy to use interval notation. Thus instead of saying that x is between 0 and π, we can use the open interval (0;π) if the endpoints are not included, or the closed interval [0;π] if the endpoints are included.
You can also have a half-open interval. For instance, the interval [0;2π) is all numbers ≥ 0 and < 2π. You could also say it’s the interval from 0 (inclusive) to 2π (exclusive).
These pages will show examples with both radians and degrees. The same theorems apply to either way of measuring an angle, and you need to practice with both.
A lot of students seem to find radians terrifying. But measuring angles in degrees and radians is no worse than measuring temperature in °F and °C. In fact, angle measure is easier because 0°F and 0°C are not the same temperature, but 0° and 0 radians are the same angle.
Just remember that a complete circle is 2π radians, and therefore half a circle is π radians:
π radians (or just π) = 180°
You can convert between degrees and radians exactly the same way you convert between inches and feet, or between centimeters and meters. (If conversions in general are a problem for you, you might like to consult my page on that topic.)
But even though you can convert between degrees and radians, it’s probably better to learn to think in both. Here’s an analogy.
When you learn a foreign language L, you go through a stage where you mentally translate what someone says in L into your own language, formulate your answer in your own language, mentally translate it into L, and then speak. Eventually you get past that stage, and you carry on a conversation in the other language without translating. Not only is it more fun, it’s a heck of a lot faster and easier.
One easy way to visualize the special angles in radians is to think of the twelve hours numbered around the circumference of a clock face. One hour of travel by the hour hand is π/6, two hours is π/3, three hours is π/2, six hours is π, and so on. (Thanks are due to Jeffrey T. Birt for this suggestion.)
You want to train yourself to work with radians for the same reason: it’s more efficient, and saving work is always good. Practice visualizing an angle of π/6 or 3π/4 or 5π/3 directly, without translating to degrees. You’ll be surprised how quickly it will become second nature!
Trig without Tears Part 2:
Summary: Every one of the six trig functions is just one side of a right triangle divided by another side. Or if you draw the triangle in a unit circle, every function is the length of one line segment. The easy way to remember all six definitions: memorize the definitions of sine and cosine and then remember the other four as combinations of sine and cosine, not as independent functions.
A picture is worth a thousand words (which is why it takes a thousand times as long to download). The trig functions are nothing more than lengths of various sides of a right triangle in various ratios. Since there are three sides, there are 3×2 = 6 different ways to make a ratio (fraction) of sides. That’s why there are six trig functions, no more and no less.
Of those six functions, three—sine, cosine, and tangent—get the lion’s share of the work. (The others are studied because they can be used to make some expressions simpler.) We’ll begin with sine and cosine, because they are really basic and the others depend on them.
Here is one of the conventional ways of showing
a right triangle. A key point is that the lower-case letters a, b, c
are the sides opposite to the angles marked with the corresponding
capital letters A, B, C. Most books use this convention:
lower-case letter for side opposite upper-case angle.
The two fundamental definitions are marked in the diagram. You must commit them to memory. In fact, they should become second nature to you, so that you recognize them no matter how the triangle is turned around. Always, always, the sine of an angle is equal to the opposite side divided by the hypotenuse (opp/hyp in the diagram). The cosine is equal to the adjacent side divided by the hypotenuse (adj/hyp).
(1) memorize:
sine = (opposite side) / hypotenuse
cosine = (adjacent side) / hypotenuse
What is the sine of B in the diagram? Remember opp/hyp: the opposite side is b and the hypotenuse is c, so sin B = b/c. What about the cosine of B? Remember adj/hyp: the adjacent side is a, so cos B = a/c.
Do you notice that the sine of one angle is the cosine of the other? Since A+B+C = 180° for any triangle, and C is 90°, A+B must equal 90°. Therefore A = 90°−B and B = 90°−A. When two angles add to 90°, each angle is the complement of the other, and the sine of each angle is the cosine of the other. These are the cofunction identities:
(2) sin A = cos(90°−A) or cos(π/2−A)
cos A = sin(90°−A) or sin(π/2−A)
The definitions of sine and cosine can be rearranged a little bit to let you write down the lengths of the sides in terms of the hypotenuse and the angles. For example, when you know that b/c = cos A, you can multiply through by c and get b = c×cos A. Can you write another expression for length b, one that uses a sine instead of a cosine? Remember that opposite over hypotenuse equals the sine, so b/c = sin B. Multiply through by c and you have b = c×sin B.
Can you see how to write down two expressions for the length of side a? Please work from the definitions and verify that a = c×sin A = c×cos B.
Example: Given a right triangle with angle A =52° and hypotenuse c = 150 m. What is the length of side b?
Solution: b = c×cos A = 150×cos 52° = about 92.35 m.
Example: A guy wire is anchored in the ground and attached to the top of a 45-foot flagpole. If it meets the ground at an angle of 63°, how long is the guy wire?
Solution: This is a right triangle, with A = 63°, a = 45 ft, and hypotenuse c unknown. But you do know that a = c×sin A, and therefore c = a/sin A = 45/sin 63° = about 50.5 ft.
You may be wondering how to find sides or angles of triangles when there is no right angle. We’ll get to that, under the topic of Solving Triangles.
One important special case comes up frequently. Suppose the hypotenuse c = 1; then we call the triangle a unit right triangle. You can see from the paragraphs just above that if c = 1 then a = sin A and b = cos A. In other words, in a unit right triangle the opposite side will equal the sine and the adjacent side will equal the cosine of the angle.
The triangle is often drawn in a unit circle, a circle of radius 1, as shown at right. The angle A is at the center and the adjacent side lies along the x axis. If you do this, the hypotenuse is the radius, which is 1. The (x,y) coordinates of the outer end of the hypotenuse are the x and y legs of the triangle (cos A,sin A). The unit circle is your friend: it can help you visualize lots of trig identities.
The other four functions have no real independent life of their own; they’re just combinations of the first two. You could do all of trigonometry without ever knowing more than sines and cosines. But knowing something about the other four, especially the tangent, can often save you some steps in a calculation—and your teacher will expect you to know about them for exams.
I find it easiest to memorize (sorry!) the definition of the tangent in terms of the sine and cosine:
(3) memorize:
tan A = (sin A) / (cos A)
You’ll use the tangent (tan) function very much more than the last three functions. (I’ll get to them in a minute.)
There’s an alternative way to remember the meaning of the tangent. Remember from the diagram that sin A = opposite/hypotenuse and cos A = adjacent/hypotenuse. Plug those into equation 3, the definition of the tan function, and you have tan A = (opposite/hypotenuse) / (adjacent/hypotenuse) or
(4) tangent = (opposite side) / (adjacent side)
Notice this is not marked “memorize”: you don’t have to memorize it because it flows directly from the definition equation 3, and in fact the two statements are equivalent. I’ve chosen to present them in this order to minimize the jumble of opp, adj, and hyp among sin, cos, and tan. However, if you prefer you can memorize equation 4 and then derive the equivalent identity equation 3 whenever you need it.
The other three trig functions—cotangent, secant, and cosecant—are defined in terms of the first three. They’re much less often used, but they do simplify some problems in calculus.
(5) memorize:
cot A = 1 / (tan A)
sec A = 1 / (cos A)
csc A = 1 / (sin A)
Guess what! That’s the last trig identity you have to memorize.
(You’ll probably find that you end up memorizing certain other identities without even intending to, just because you use them frequently. But equation 5 makes the last ones that you’ll have to sit down and make a point of memorizing just on their own.)
Unfortunately, the definitions in equation 5 aren’t the easiest thing in the world to remember. Does the secant equal 1 over the sine or 1 over the cosine? Here are two helpful hints: Each of those definitions has a co-function on one and only one side of the equation, so you won’t be tempted to think that sec A equals 1/sin A. And secant and cosecant go together just like sine and cosine, so you won’t be tempted to think that cot A equals 1/sin A.
For an alternative approach to remembering the above identities, you might like this 24-second video, suggested by Gene Laratonda: <http://www.youtube.com/watch?v=McIbWHgjQng>
You can immediately notice an important relation between tangent and cotangent. Each is the co-function of the other, just like sine and cosine:
(6) tan A = cot(90°−A) or cot(π/2−A)
cot A = tan(90°−A) or tan(π/2−A)
If you want to prove this, take the definition of tan and use equation 2 to substitute cos(90°−A) for sin A and sin(90°−A) for cos A. Tangent and cotangent are co-functions just like sine and cosine. By doing the same sort of substitution, you can show that secant and cosecant are also co-functions:
(7) sec A = csc(90°−A) or csc(π/2−A)
csc A = sec(90°−A) or sec(π/2−A)
The formulas for negative angles of tangent and the other functions drop right out of the definitions equation 3 and equation 5, since you already know the formulas equation 22 for sine and cosine of negative angles. For instance, tan(−A) = sin(−A)/cos(−A) = −sin A/cos A.
(8) tan(−A) = −tan A
cot(−A) = −cot A
sec(−A) = sec A
csc(−A) = −csc A
You can reason out things like whether sec(180°−A) equals sec A or −sec A: just apply the definition and use what you already know about cos(180°−A).
You saw earlier how the sine and cosine of an angle are the sides of a triangle in a unit circle. It turns out that all six functions can be shown geometrically in this way.
unit circle (radius = AB = 1)
sin θ = BC;
cos θ = AC;
tan θ = ED
csc θ = AG;
sec θ = AE;
cot θ = FG
graphic courtesy of
TheMathPage
In the illustration at right, triangle ABC has angle θ at the center of a unit circle (AB = radius = 1). You already know that BC = sin θ and AC = cos θ.
What about tan θ? Well, since DE is tangent to the unit circle, you might guess that its length is tan θ, and you’d be right. Triangles ABC and AED are similar, and therefore
ED / AD = BC / AC
ED / 1 = sin θ / cos θ
ED = tan θ
More information comes from the same pair of similar triangles:
AE / AB = AD / AC
AE / 1 = 1 / cos θ
AE = sec θ
The lengths that represent cot θ and csc θ will come from the other triangle, GAF. That triangle is also similar to triangle AED. (Why? FG is perpendicular to FA, and FA is perpendicular to AD; therefore FG and AD are parallel. In beginning geometry you learned that when parallel lines are cut by a third line, the corresponding angles—marked θ in the diagram—are equal. Thus FG is a tangent to the unit circle, and therefore angles G and θ are equal. )
Using similar triangles GAF and AED,
FG / FA = AD / ED
FG / 1 = 1 / tan θ
FG = cot θ
That makes sense: FG is tangent to the unit circle, and is the tangent of the complement of angle θ, namely angle GAF, and therefore it is the cotangent of the original angle θ (or angle GAD).
Finally, using the same pair of similar triangles again, you can also say that
AG / FA = AE / ED
AG / 1 = sec θ / tan θ
AG = [ 1 / cos θ ] / [ sin θ / cos θ ]
AG = 1 / sin θ
AG = csc θ
This one diagram beautifully depicts the geometrical meaning of all six trig functions when the angle θ is drawn at the center of a unit circle:
sin θ = BC; cos θ = AC; tan θ = ED
csc θ = AG; sec θ = AE; cot θ = FG
From the picture, it’s obvious why the name “tangent” makes sense: the tangent of an angle is the length of a segment tangent to the unit circle. But what about the sine function? How did it get its name?
Please look at the picture again, and notice that sin θ = BC is half a chord of the circle. The Hindu mathematician Aryabhata the elder (about 475-550) used the word “jya” or “jiva” for this half-chord. In Arabic translation the word was unchanged, but in the Arabic system of writing “jiva” was written the same way as the Arabic word “jaib”, meaning bosom, fold, or bay. The Latin word for bosom, bay, or curve is “sinus”, or “sine” in English, and beginning with Gherardo of Cremona (about 1114-1187) that became the standard term.
Edmund Gunter (1581-1626) seems to have been the first to publish the abbreviations sin and tan for sin and tangent.
My source for this history is Eli Maor’s Trigonometric Delights (1998, Princeton University Press), pages 35-36. I urge you to consult the book for a fuller account.
Trig without Tears Part 3:
Summary: You need to know the function values of certain special angles, namely 30° (π/6), 45° (π/4), and 60° (π/3). You also need to be able to go backward and know what angle has a sine of ½ or a tangent of −√3. While it’s easy to work them out as you go (using easy right triangles), you really need to memorize them because you’ll use them so often that deriving them or looking them up every time would really slow you down.
Look at this 45-45-90° triangle, which
means sides a and b are equal. By the Pythagorean theorem,
a² + b² = c²
But a = b and c = 1; therefore
2a² = 1
a² = 1/2
a = 1/√2 = (√2)/2
Since a = sin 45°,
sin 45° = (√2)/2
Also, b = cos 45° and b = a; therefore
cos 45° = (√2)/2
Use the definition of tan A, equation 3 or equation 4:
tan 45° = a/b = 1
(14) sin 45° = cos 45° = (√2)/2
tan 45° = 1
Now look at this diagram. I’ve drawn two 30-60-90° triangles back
to back, so that the two 30° angles are next to each other.
Since 2×30° = 60°, the big triangle is a 60-60-60°
equilateral triangle. Each of the small triangles has hypotenuse 1, so
the length 2b is also 1, which means that
b = ½2s
But b also equals cos 60°, and therefore
cos 60° = ½
You can find a, which is sin 60°, by using the Pythagorean theorem:
(½)² + a² = c² = 1
1/4 + a² = 1
a² = 3/4 ⇒ a = (√3)/2
Since a = sin 60°, sin 60° = (√3)/2.
Since you know the sine and cosine of 60°, you can easily use the cofunction identities (equation 2) to get the cosine and sine of 30°:
cos 30° = sin(90°−30°) = sin 60° = (√3)/2
sin 30° = cos(90°−30°) = cos 60° = 1/2
As before, use the definition of the tangent to find the tangents of 30° and 60° from the sines and cosines:
tan 30° = sin 30° / cos 30°
tan 30° = (1/2) / ((√3)/2)
tan 30° = 1 / √3 = (√3)/3
and
tan 60° = sin 60° / cos 60°
tan 60° = ((√3)/2) / (1/2)
tan 60° = √3
The values of the trig functions of 30° and 60° can be summarized like this:
(15) sin 30° = ½, sin 60° = (√3)/2
cos 30° = (√3)/2, cos 60° = ½
tan 30° = (√3)/3, tan 60° = √3
Incidentally, the sines and cosines of 0, 30°, 45°, 60° and 90° display a pleasing pattern:
(16) for angle A = 0, 30° (π/6), 45° (π/4), 60° (π/3), 90° (π/2):
sin A = (√0)/2, (√1)/2, (√2)/2, (√3)/2, (√4)/2
cos A = (√4)/2, (√3)/2, (√2)/2, (√1)/2, (√0)/2
tan A = 0, (√3)/3, 1, √3, undefined
It’s not surprising that the cosine pattern is a mirror image of the sine pattern, since sin(90°−A) = cos A.
Trig without Tears Part 4:
Summary: The six trig functions were originally defined for acute angles in triangles, but now we define them for any angle (or any number). If you want any of the six function values for an angle that’s not between 0 and 90° (π/2), you just find the function value for the reference angle that is within that interval, and then possibly apply a minus sign.
So far we have defined the
six trig functions as ratios of
sides of a right triangle. In a right triangle, the other two angles
must be less than 90°, as suggested by the picture at left.
Suppose you draw the triangle in a circle this way, with angle A at the origin and the circle’s radius equal to the hypotenuse of the triangle. The hypotenuse ends at the point on the circle with coordinates (x,y), where x and y are the lengths of the two legs of the triangle. Then using the standard definitions of the trig functions, you have
sin A = opposite/hypotenuse = y/r
cos A = adjacent/hypotenuse = x/r
This is the key to extending the trig functions to any angle.
The trig functions had their roots in measuring sides of triangles,
and chords of a circle (which is practically the same thing). If we
think about an angle in a circle, we can extend the trig functions
to work for any angle.
In the diagram, the general angle A is drawn in standard position, just as we did above for an acute angle. Just as before, its vertex is at the origin and its initial side lies along the positive x axis. The point where the terminal side of the angle cuts the circle is labeled (x,y).
(This particular angle happens to be between 90° and 180° (π/2 and π), and we say it lies in Quadrant II. But you could draw a similar diagram for any angle, even a negative angle or one >360°.)
Now let’s define sine and cosine of angle A, in terms of the coordinates (x,y) and the radius r of the circle:
(21) sin A = y/r, cos A = x/r
This is nothing new. As you saw above when A was in Quadrant I, this is exactly the definition you already know from equation 1: sin A = opposite/hypotenuse, cos A = adjacent/hypotenuse. We’re just extending it to work for any angle.
The other function definitions don’t change at all. From equation 3 we still have
tan A = sin A / cos A
which means that
tan A = y/x
and the other three functions are still defined as reciprocals (equation 5).
Once again, there’s nothing new here: we’ve just extended the original definitions to a larger domain.
So why go through this? Well, for openers, not every triangle is an acute triangle. Some have an angle greater than 90°. Even in down-to-earth physical triangles, you’ll have to be concerned with functions of angles greater than 90°.
Beyond that, it turns out that all kinds of physical processes vary in terms of sines and cosines as functions of time: height of the tide; length of the day over the course of a year; vibrations of a spring, or of atoms, or of electrons in atoms; voltage and current in an AC circuit; pressure of sound waves, Nearly every periodic process can be described in terms of sines and cosines.
And that leads to a subtle shift of emphasis. You started out thinking of trig functions of angles, but really the domain of trig functions is all real numbers, just like most other functions. How can this be? Well, when you think of an “angle” of so-and-so many radians, actually that’s just a pure number. For instance, 30°=π/6. We customarily say “radians” just to distinguish from degrees, but really π/6 is a pure number. When you take sin(π/6), you’re actually evaluating the function sin(x) at x = π/6 (about 0.52), even though traditionally you’re taught to think of π/6 as an angle.
We won’t get too far into that in these pages, but here’s an example. If the average water depth is 8 ft in a certain harbor, and the tide varies by ±3 ft, then the height at time t is given by a function that resembles y = 8 + 3 cos(0.52t). (It’s actually more complicated, because high tides don’t come at the same time every day, but that’s the idea.)
Coming back from philosophy to the nitty-gritty of computation, how do we find the value of a function when the angle (or number) is outside the range [0;90°] (which is 0 to π/2)? The key is to define a reference angle.
Here’s the same picture of angle A again, but with its reference angle
added. With angle A in standard position, the reference angle is the
acute angle between the terminal side of A and the
positive or negative x axis.
In this case, angle A is in Q II, the reference angle is
180°−A (π−A). Why? Because the two angles together
equal 180° (π).
What good does the reference angle do you? Simply this: the six function values for any angle equal the function values for its reference angle, give or take a minus sign.
That’s an incredibly powerful statement, if you think about it. In the drawing, A is about 150° and the reference angle is therefore about 30°. Let’s say they’re exactly 150° and 30°, just for discussion. Then sine, cosine, tangent, cotangent, secant, and cosecant of 150° are equal to those same functions of 30°, give or take a minus sign.
What’s this “give or take” business? That’s what the next section is about.
Remember the extended definitions from equation 21:
sin A = y/r, cos A = x/r
The radius r is always taken as positive, and therefore the signs of sine and cosine are the same as the signs of y and x. But you know which quadrants have positive or negative y and x, so you know for which angles (or numbers) the sine and cosine are positive or negative. And since the other functions are defined in terms of the sine and cosine, you also know where they are positive or negative.
Spend a few minutes thinking about it, and draw some sketches. For instance, is cos 300° positive or negative? Answer: 300° is in Q IV, which is in the right-hand half of the circle. Therefore x is positive, and the cosine must be positive as well. The reference angle is 60° (draw it!), so cos 300° equals cos 60° and not −cos 60°.
You can check your thinking against the chart that follows. Whatever you do, don’t memorize the chart! Its purpose is to show you how to reason out the signs of the function values whenever you need them, not to make you waste storage space in your brain.
| Signs of Function Values | ||||
|---|---|---|---|---|
| Q I 0 to 90° 0 to π/2 |
Q II 90 to 180° π/2 to π |
Q III 180 to 270° π to 3π/2 |
Q IV 270 to 360° 3π/2 to 2π |
|
| x | positive | negative | negative | positive |
| y | positive | positive | negative | negative |
| sin A (= y/r) |
positive | positive | negative | negative |
| cos A (= x/r) |
positive | negative | negative | positive |
| tan A (= y/x) |
positive | negative | positive | negative |
What about other angles? Well, 420° = 360°+60°, and therefore 420° ends in the same position in the circle as 60°—it’s just going once around the circle and then an additional 60°. So 420° is in Q I, just like 60°.
You can analyze negative angles the same way. Take −45°. That occupies the same place on the circle as +315° (360°−45°). −45° is in Q IV.
As you’ve seen, for any function you get the numeric value by considering the reference angle and the positive or negative sign by looking where the angle is.
Example: What’s cos 240°? Solution: Draw the angle and see that the reference angle is 60°; remember that the reference angle always goes to the x axis, even if the y axis is closer. cos 60° = ½, and therefore cos 240° will be ½, give or take a minus sign. The angle is in Q III, where x is negative, and therefore cos 240° is negative. Answer: cos 240° = −½.
Example: What’s tan(−225°)? Solution: Draw the angle and find the reference angle of 45°. tan 45° = 1. But −225° is in Q II, where x is negative and y is positive; therefore y/x is negative. Answer: tan(−225°) = −1.
The techniques we worked out above can be generalized into a set of identities. For instance, if two angles are supplements then you can write one as A and the other as 180°−A. You know that one will be in Q I and the other in Q II, and you also know that one will be the reference angle of the other. Therefore you know at once that the sines of the two angles will be equal, and the cosines of the two will be numerically equal but have opposite signs.
This diagram may help:
Here you see a unit circle (r = 1) with four identical triangles. Their angles A are at the origin, arranged so that they’re mirror images of each other, and their hypotenuses form radii of the unit circle. Look at the triangle in Quadrant I. Since its hypotenuse is 1, its other two sides are cos A and sin A.
The other three triangles are the same size as the first so their sides must be the same length as the sides of the first triangle. But you can also look at the other three radii as belonging to angles 180°−A in Quadrant II, 180°+A in Quadrant III, and −A or 360°−A in Quadrant IV. All the others have a reference angle equal to A. From the symmetry, you can immediately see things like sin(180°+A) = −sin A and cos(−A) = cos A.
The relations are summarized below. Don’t memorize them! Just draw a diagram whenever you need them—it’s easiest if you use a hypotenuse of 1. Soon you’ll find that you can quickly visualize the triangles in your mind and you won’t even need to draw a diagram. The identities for tangent are easy to derive: just divide sine by cosine as usual.
| sin(180°−A) = sin A
sin(π−A) = sin A |
cos(180°−A) = −cos A
cos(π−A) = −cos A |
tan(180°−A) = −tan A
tan(π−A) = −tan A |
(22) |
| sin(180°+A) = −sin A
sin(π+A) = −sin A |
cos(180°+A) = −cos A
cos(π+A) = −cos A |
tan(180°+A) = tan A
tan(π+A) = tan A |
|
| sin(−A) = −sin A | cos(−A) = cos A | tan(−A) = −tan A |
You should be able to see that 360° brings you all the way around the circle. That means that an angle of 360°+A or 2π+A is the same as angle A. Therefore the function values are unchanged when you add 360° or a multiple of 360° (or 2π or a multiple) to the angle. Also, if you move in the opposite direction for angle A, that’s the same angle as 360°−A or 2π−A, so the function values of −A and 360°−A (or 2π−A) are the same.
For this reason we say that sine and cosine are periodic functions with a period of 360° or 2π. Their values repeat over and over again. Of course secant and cosecant, being reciprocals of cosine and sine, must have the same period.
What about tangent and cotangent? They are periodic too, but their period is 180° or π: they repeat twice as fast as the others. You can see this from equation 22: tan(180°+A) = tan A says that the function values repeat every 180°.
Trig without Tears Part 5:
Summary: A triangle has six parts, three sides and three angles. Given almost any three of them—three sides, two sides and an angle, or one side and two angles—you can find the other three values. This is called solving the triangle, and you can do it with any triangle, not just a right triangle.
Let’s look at a specific example to start with. Suppose you have a triangle where one side has a length of 180, an adjacent angle is 42°, and the opposite angle is 31°. You’re asked to find the other angle and the other two sides.
It’s always a good idea to draw a rough sketch, like this one. Not
only does it help you organize your solution process better, but it
can help you check your work. For instance, since the 31° angle
is the smallest, you know that the opposite side must also be the
shortest. If you were to come up with an answer of, say, 110 for one
of the other sides, you’d know at once that you had made a mistake
somewhere because 110 is < 180 and the other two sides must
both be > 180.
How would you go about solving this problem? It’s not immediately obvious, I agree. But maybe we can get some help from some useful general techniques in problem solving:
We’ve already got the diagram, but let’s see if those other techniques will be helpful. (By the way, they’re not original with me, but are from a terrific book on problem-solving techniques that I think you should know about.)
“Can you use what you already know to solve a piece of this problem?” For example, if this were a right triangle you’d know right away how to write down the lengths of sides in terms of sines or cosines.
But it’s not a right triangle, alas. Is there any way to turn it into a right triangle? Not exactly, but if you construct a line at right angles to one side and passing through the opposite vertex, you’ll have two right triangles. Maybe solving those right triangles will show how to solve the original triangle.
This diagram shows the same triangle after I drew that
perpendicular. I’ve also used another principle (“Can you solve a more
general problem?”) and replaced the specific numbers with the usual
letters for sides and angles. Dropping perpendicular CD in the
diagram divides the big triangle (which you don’t know how to
solve) into two right triangles ACD and BCD, with a common side CD.
And you can solve those right triangles.
We’re going to use this simple diagram to develop two important tools for solving triangles: the Law of Sines and the Law of Cosines. Just drawing this one perpendicular line will show you how to solve not just the triangle we started with, but any triangle. (Some trig courses teach other laws like the Law of Tangents and the Law of Segments. I’m ignoring them because you can solve triangles just fine without them.)
The Law of Sines is simple and beautiful and easy to derive. It’s useful when you know two angles and any side of a triangle, or sometimes when you know two sides and one angle.
Let’s start by writing down things we know that relate the sides and angles of the two right triangles in the diagram above. You remember how to write down the lengths of the legs of a right triangle? The leg is always equal to the hypotenuse times either the cosine of the adjacent angle or the sine of the opposite angle. (If that looks like just empty words to you, or even if you’re not 100% confident about it, please go back and review that section until you feel confident.)
In the diagram above, look at triangle ADC at the left: the right angle is at D and the hypotenuse is b. We don’t know how much of original angle C is in this triangle, so we can’t use C to find the lengths of any sides. What can we write down using angle A? By using its cosine and sine we can write the lengths of both legs of the triangle:
AD = b×cos A and CD = b×sin A
By the same reasoning, in the other triangle you have
DB = a×cos B and CD = a×sin B
This is striking: you see two different expressions for the length CD. But things that are equal to the same thing are equal to each other. That means that
b×sin A = a×sin B
Divide through by sin A and you have the solution:
b = a×sin B / sin A
b = 180 ×sin 42° / sin 31° = about 234
What about the third angle, C, and the third side, c? Well, when you have two angles of a triangle you can find the third one easily:
A+B+C = 180°
C = 180°−A−B
In this case, C = 180−31−42° = 107°.
For the third side, there are a couple of ways to go. You wrote expressions above for AD and DB, and you know that c = AD+DB, so you could compute c = b×cos A + a×cos B.
But that’s two multiplies and an add, a bit more complicated than the one multiply and one divide to find side b. I’m lazy, and I like to reduce the amount of tapping I do on my calculator. Is there an easier way, even if just slightly easier? Yes, there is. Go back a step, to
a×sin B = b×sin A
Divide left and right by (sin A)(sin B) to get
a / sin A = b / sin B
But there’s nothing special about the two angles A and B. You could
just as well have dropped a perpendicular from A to BC or from B to
AC. Shown at right is the result of dropping a perpendicular from B to
line CD.
Because C > 90°, this perpendicular happens to be outside the triangle and the two right triangles ABD and CBD overlap. But this won’t affect the algebra. By the way, the angle in triangle CBD is not C but 180°−C, the supplement of C. Angle C belongs to the original triangle ABC.
You can write the length of the common side BD as
BD = c×sin A (in triangle ABD)
and
BD = a×sin(180−C) (in triangle CBD)
But sin(180−C) = sin C, so you have
BD = a×sin C (in triangle CBD)
Set the two computed lengths of BD equal to each other, and divide by (sin A)(sin C):
a×sin C = c×sin A
a / sin A = c / sin C
But we already figured out earlier that
a / sin A = b / sin B
Combining these two equations you have the Law of Sines:
a / sin A = b / sin B = c / sin C
It’s nice that the derivation doesn’t make you worry about obtuse versus acute triangles. As you see, when an obtuse angle is involved some dropped perpendicular will lie outside the original triangle, and in that case the derivation uses 180° minus an angle of the original triangle. But since sin x = sin(180°−x), you end up with the same form of the Law of Sines whether the perpendicular is inside or outside the triangle, whether all three angles are acute or one is obtuse.
(28) Law of Sines—First Form:
a / sin A = b / sin B = c / sin C
This is very simple and beautiful: for any triangle, if you divide any of the three sides by the sine of the opposite angle, you’ll get the same result. This law is valid for any triangle.
The Law of Sines is sometimes given upside down:
(29) Law of Sines—Second Form:
sin A / a = sin B / b = sin C / c
Of course that’s the same law, just as 2/3 = 6/9 and 3/2 = 9/6 are the same statement. Work with it either way and you’ll come up with the same answers.
You can derive the Law of Sines at need, so I don’t specifically recommend memorizing it. But it’s so simple and beautiful that it’s pretty hard not to memorize if you use it at all. It’s also pretty hard to remember it wrong: there are no alternating plus and minus signs or combinations of different functions.
In most cases where you use the Law of Sines, you get a unique solution. But sometimes you get two solutions (or none) in the side-side-angle case, where you know two sides and an angle that’s not between them. Please see the Special Note below, after the table.
The Law of Sines is fine when you can relate sides and angles. But suppose you know three sides of the triangle—for instance a = 180, b = 238, c = 340—and you have to find the three angles. The Law of Sines is no good for that because it relates two sides and their opposite angles. If you don’t know any angles, you have an equation with two unknowns and you can’t solve it.
But a triangle can be solved when you know all three sides;
you just need a different tool. And knowing me, you can be sure I’m
going to help you develop one! It’s called the Law of Cosines.
Let’s look back at that generic triangle with a perpendicular dropped from vertex C. You may remember that when we first looked at this picture, we pulled out information using both the sine and the cosine of the two angles. We used the sine information to develop the Law of Sines, but we never went anywhere with the cosine information, which was
AD = b cos A and DB or BD = a cos B
Let’s see where that can lead us. You remember that the way we came up with the Law of Sines was to write two equations that featured the length of the construction line CD, and then combine the equations to eliminate CD. Can we do anything like that here?
Well, we know the other two sides of those right triangles, so we can write an expression for the height CD using the Pythagorean theorem—actually, two expressions, one for each triangle.
a² = (CD)² + (BD)² ⇒ (CD)² = a² − (BD)²
b² = (CD)² + (AD)² ⇒ (CD)² = b² − (AD)²
and therefore
a² − (BD)² = b² − (AD)²
Substitute the known values of BD and AD in terms of angles and sides of the original triangle, and you have
a² − a²cos²B = b² − b²cos²A
Bzzt! No good! That uses two sides and two angles, but we need an equation in three sides and one angle, so that we can solve for that angle. Let’s back up a step, to a² − (BD)² = b² − (AD)², and see if we can go in a different direction.
Maybe the problem is in treating BD and AD as separate entities when actually they’re parts of the same line. Since BD+AD = c, we can write
BD = c−AD = c − b×cos A.
Notice that this brings in the third side, c, and angle B drops out. Substituting, we now have
a² − (BD)² = b² − (AD)²
a² − (c − b×cos A)² = b² − (b×cos A)²
This looks worse than the other one, but actually it’s better because it’s what we’re looking for: an equation for the three sides and one angle. We can solve it with a little algebra:
a² − c² + 2bc cos A − b²cos²A = b² − b²cos²A
a² − c² + 2bc cos A = b²
2bc cos A = b² + c² − a²
cos A = (b² + c² − a²) / 2bc
We were a long time getting there, but finally we made it. Now
we can plug in the lengths of the sides and come up with a value for
cos A, which in turn will tell us angle A:
cos A = (238² + 340² − 180²) / (2×238×340)
cos A = 0.864088
A = about 30.2°
Do the same thing to find the second angle (or use the Law of Sines, since it’s less work), then subtract the two known angles from 180° to find the third angle.
You can find the Law of Cosines for the other angles by following the same process using the other two perpendiculars.
(30) Law of Cosines—First Form:
cos A = (b² + c² − a²) / 2bc
cos B = (a² + c² − b²) / 2ac
cos C = (a² + b² − c²) / 2ab
Just for fun, let’s find angle C for that triangle:
cos C = (a² + b² − c²) / 2ab
cos C = (180² + 238² − 340²) / 2×180×238
cos C = −0.309944
Don’t be surprised at the negative number. Remember from the diagram in Functions of Any Angle that cos A < 0 when A is between 90° and 180°. Because the cosine has unique values all the way from 0° to 180°, you never have to worry about multiple solutions of a triangle when you use the Law of Cosines. In this case, C is about 108°.
There’s another well-known form of the Law of Cosines, which may be a bit easier to remember. Start with the above form, multiply through by 2ab, and isolate c on one side:
cos C = (a² + b² − c²) / 2ab
2ab×cos C = a² + b² − c²
c² = a² + b² − 2ab×cos C
Notice the arrangement: side c is opposite angle C in the triangle, and they’re at opposite ends of this equation. Sides a and b are adjacent to angle C both in the triangle and in the equation. And you can play the same game to solve for the other two sides:
(31) Law of Cosines—Second Form:
a² = b² + c² − 2bc cos A
b² = a² + c² − 2ac cos B
c² = a² + b² − 2ab cos C
Depending on how you’re using it, you may need the Law of Cosines in either of the two forms that we’ve obtained, the first form for finding an angle and the second form for finding a side.
With just the definitions of sine, cosine, and tangent, you can solve any right triangle. If you’ve got the Law of Sines and the Law of Cosines under your belt, you can solve any triangle that exists. (Some sets of givens lead to an impossible situation, like a “triangle” with sides 3-4-9.)
In this section I’ll run down the various possibilities and give you some pointers. But really it’s pretty straightforward: whenever you have to solve a triangle, think about what you have and then think about which formula you can use to get what you need. (When you have two angles, you can always find the third by A+B+C = 180°.)
Many people find it easier to think about the known elements of a triangle as a “case”. For instance, if you know two angles and the side between them, that’s case ASA; if you know two angles and a side that’s not between them, that’s case AAS, and so on.
I’m not presenting the following table for you to memorize. Instead, what I hope to do is show you that between the Law of Sines and the Law of Cosines you can solve any triangle, and that you simply pick which law to use based on which one has just one unknown and otherwise uses information you already have.
Most cases can be solved with the Law of Sines. But if you have three sides (SSS), or two sides and the angle between them (SAS), you must begin with the Law of Cosines.
The table is just an exhaustive elaboration of those two principles, so you probably don’t even need to read it! <grin>
| If you know this... | You can solve the triangle this way... | |
|---|---|---|
| three angles | There’s not enough information. Without at least one side you have the shape of the triangle, but no way to scale it correctly. For example, the same angles could give you a triangle with sides 7-12-13, 35-60-65, or any other multiple. | |
| two angles and a side, AAS or ASA | Find the third angle by subtracting from 180°. then use the Law of Sines* (equation 28) twice to find the second and third sides. | |
| two sides and ... | the included angle, SAS | Use the Law of Cosines* (equation 31) to find the third side. Then use either the Law of Sines* (equation 29) or the Law of Cosines* (equation 30) to find the second angle. |
| a non-included angle, SSA | Use the Law of Sines*
twice, equation 29 to get the second angle and
equation 28 to get the third side.
But...
|
|
| three sides, SSS | With the first form equation 30 of the Law of Cosines you use all the sides to compute one angle. Use that angle and its opposite side in the Law of Sines equation 29 to find the second angle. | |
| * If a 90° angle is given, the Law of Sines and the Law of Cosines are overkill. Just apply the definitions of the sine and cosine (equation 1) and the tangent (equation 4) to find the other sides and angles. | ||
For most sets of facts, either there’s a unique solution
or they’re obviously absurd. (If you don’t see why a
“triangle” with sides 50-60-200 is
absurd, try to sketch it.) But
the SSA case can be tricky.
Suppose you know acute angle B and sides a and b. Given those facts, there are two different ways you could draw the triangle, as shown in the picture. How can this be? Well, you use the Law of Sines to find the sines of angles A and C. Let’s say you find sin C = 0.5. That means C could be either 30° or the supplement, 150°. Remember that the sine of any angle and the sine of its supplement are the same.
This is the infamous ambiguous case. You can see the problem from the picture: the known opposite side b can take either of two positions that satisfy the given the lengths of a and b. Those two positions give rise to two different values for angle A, two different values for angle C, and two different values for side c. Think about it for a while, and you’ll see that this ambiguity can arise only when the known angle is acute, and the adjacent side is longer than the opposite side, and the opposite side is greater than the height.
Here’s a complete rundown of all the possibilities with the SSA case:
| Possibilities within the SSA Case | ||
|---|---|---|
| known angle < 90° | known angle ≥ 90° | |
| adjacent side < opposite side | one solution | one solution |
| adjacent side = opposite side | one solution | no solution (Angles that are opposite equal sides must be equal, but you can’t have two angles both ≥ 90° in a triangle.) |
| adjacent side > opposite side | Compute the triangle height h (adjacent side times sine of known
angle).
|
no solution (The conditions violate the theorem that the longest side is always opposite the largest angle.) |
For heaven’s sake, don’t try to memorize that table! Instead, always draw a picture. If you can draw two pictures that both fit all the available facts, you have two legitimate solutions. If only one picture fits all the facts, it will show you which angle (if any) is > 90°. And if you can’t make any picture that fits the facts, the triangle has no solution.
If you do have two solutions, what do you do? If you have no other information to go on, of course you report both solutions. But check the situation carefully. Maybe you’re told explicitly which is the largest angle, or it’s implied by other facts you know. In that case your solution is constrained, and you reject the solution that doesn’t meet the constraints.
I’ve written a TI-83/84 program to solve all types of triangles (and find their area). If you have TI Graph Link software, you can download the program, unzip it, and install it on your TI-83/84.
For those who don’t have the Graph Link software, here is the
program in ASCII form. Special keys or menu selections are indicated
between backslashes; for instance \->\ means the
STO key. (This is the TI-83/84 export format.)
Naturally, I’d be happy to know of any corrections or improvements to this program.
\start83P\comment=Program file dated 09/16/02, 00:23
\protected=FALSE
\name=TRIANGLE
\file=H:\TRIANGLE.TXT
Degree
Menu("WHICH CASE?","AAS",1,"ASA",2,"SAS",3,"SSA",4,"SSS",5)
Lbl 1
Disp "NEED <A,<B,SA"
Input "<A=",A
Input "<B=",B
Input "SA=",D
180-A-B\->\C
Dsin(B)/sin(A)\->\E
Dsin(C)/sin(A)\->\F
Goto 0
Lbl 2
Disp "NEED <A,SB,<C"
Input "<A=",A
Input "SB=",E
Input "<C=",C
180-A-C\->\B
Esin(A)/sin(B)\->\D
Esin(C)/sin(B)\->\F
Goto 0
Lbl 3
Disp "NEED <A,SB,SC"
Input "<A=",A
Input "SB=",E
Input "SC=",F
\root\(E\^2\+F\^2\-2EFcos(A))\->\D
sin\^-1\(Esin(A)/D)\->\B
180-A-B\->\C
Goto 0
Lbl 4
Disp "NEED SA,SB,<A"
Input "SA=",D
Input "SB=",E
Input "<A=",A
Esin(A)\->\H
If (D<H)+(A\>=\180)(D\<=\E):Then
Disp "IMPOSS":Stop
End
If (H<D)(D<E):Then
sin\^-1\(Esin(A)/D)\->\B
Menu("AMBIG--IS B","ACUTE",41,"OBTUSE",42)
Lbl 42:180-B\->\B
Goto 41
End
sin\^-1\(Esin(A)/D)\->\B
Lbl 41
180-A-B\->\C
D*sin(C)/sin(A)\->\F
Goto 0
Lbl 5
Input "SA=",D
Input "SB=",E
Input "SC=",F
cos\^-1\((E\^2\+F\^2\-D\^2\)/(2EF))\->\A
cos\^-1\((D\^2\+F\^2\-E\^2\)/(2DF))\->\B
180-A-B\->\C
Lbl 0
.5DEsin(C)\->\G
Disp "AREA <S SIDES",round(G,2),{round(A,1),round(B,1),
round(C,1)},round(D,2),round(E,2),round(F,2)
Caution: that long DISP command at the end should be entered as a single command; I’ve just broken it into two lines for readability.
Trig without Tears Part 6:
Summary: This chapter begins exploring trigonometric identities. Three of them involve only squares of functions. These are called Pythagorean identities because they’re just the good old Theorem of Pythagoras in new clothes. Learn the really basic one, namely sin²A + cos²A = 1, and the others are easy to derive from it in a single step.
Students seem to get bogged down in the huge number of trigonometric identities. As I said earlier, I think the problem is that students are expected to memorize all of them. But really you don’t have to, because they’re all just forms of a very few basic identities. The next couple of chapters will explore that idea.
For example, let’s start with the really basic identity:
(38) sin²A + cos²A = 1
That one’s easy to remember: it involves only the basic sine and cosine, and you can’t get the order wrong unless you try.
But you don’t have to remember even that one, since it’s really just another
form of the Pythagorean theorem. (You do remember that, I
hope?) Just think about a right triangle with a hypotenuse of one
unit, as shown at right.
First convince yourself that the figure is right, that the lengths of the two legs are sin A and cos A. (Check back in the section on lengths of sides, if you need to.) Now write down the Pythagorean theorem for this triangle. Voilà! You’ve got equation 38.
What’s nice is that you can get the other “squared” or Pythagorean identities from this one, and you don’t have to memorize any of them. Just start with equation 38 and divide through by either sin²A or cos²A.
For example, what about the riddle we started with, the relation between tan²A and sec²A? It’s easy to answer by a quick derivation—easier than memorizing, in my opinion.
If you want an identity involving tan²A, remember equation 3: tan A is defined to be sin A/cos A. Therefore, to create an identity involving tan²A you need sin²A/cos²A. So take equation 38 and divide through by cos²A:
sin²A + cos²A = 1
sin²A/cos²A + cos²A/cos²A = 1/cos²A
(sin A/cos A)² + 1 = (1/cos A)²
which leads immediately to the final form:
(39) tan²A + 1 = sec²A
You should be able to work out the third identity (involving cot²A and csc²A) easily enough. You can either start with equation 39 above and use the co-function rules equation 6 and equation 7, or start with equation 38 and divide by something appropriate. Either way, check to make sure that you get
(40) cot²A + 1 = csc²A
It may be easier for you to visualize these identities geometrically.
In the diagram at right, remember that we showed that ED = tan A and AE = sec A. AD is a radius of the unit circle and therefore AD = 1. Since angle D is a right angle, you can use the Pythagorean theorem:
(ED)² + (AD)² = (AE)²
tan²A + 1 = sec²A
You can use triangle AFG in a similar way to prove equation 40.
Trig without Tears Part 7:
Summary: Continuing with trig identities, this page looks at the sum and difference formulas, namely sin(A± B), cos(A± B), and tan(A± B). Remember one and all the rest flow from it. There’s also a beautiful way to get them from Euler’s formula.
Formulas for cos(A+B), sin(A−B), and so on are important but hard to remember. Yes, you can derive them by strictly trigonometric means. But such proofs are lengthy, too hard to reproduce when you’re in the middle of an exam or of some long calculation.
This brings us to W.W. Sawyer’s marvelous idea, as expressed in chapter 15 of Mathematician’s Delight (1943; reprinted 1991 by Penguin Books). He shows how you can derive the sum and difference formulas by ordinary algebra and one simple formula.
The ordinary algebra is simply the rules for combining powers:
(46) xaxb = xa+b
(xa)b = xab
(If you’re a bit rusty on the laws of exponents, you may want to review them.)
You may already know the “simple formula” that I mentioned above. It’s
(47) memorize:
cos x + i sin x = eix
The formula is not Sawyer’s, by the way; it’s commonly called Euler’s formula. I don’t even know whether the idea of using Euler’s formula to get the sine and cosine of sum and difference is original with Sawyer. But I’m going to give him credit, since his explanation is simple and clear and I’ve never seen it explained in this way anywhere else.
You’ll sometimes see cos x + i sin x abbreviated as cis x for brevity.
I’ve marked Euler’s formula, equation 47, “memorize”. Although it’s not hard to derive (and Sawyer does it in a few steps by means of power series), you have to start somewhere. And that formula has so many other applications that it’s well worth committing to memory. For instance, you can use it to get the roots of a complex number and the logarithm of a negative number.
Okay, back to Sawyer’s idea. What happens if you substitute x = A+B in equation 47 above? You get
cos(A+B) + i sin(A+B) = e(iA+iB)
Hmmm, this looks interesting. It involves exactly what we’re looking for, cos(A+B) and sin(A+B). Can you simplify the right-hand side? Use equation 46 and then equation 47 to rewrite it:
eiA+iB = eiA eiB = (cos A + i sin A)(cos B + i sin B)
Now multiply that out and set it equal to the original left-hand side:
cos(A+B) + i sin(A+B) = [cos A cos B − sin A sin B] + i[sin A cos B + cos A sin B]
Now here’s the sneaky part. If I told you a+bi = 7−9i and asked you to solve for a and b, you could immediately tell me that a = 7 and b = −9, right? More formally, if two complex numbers are equal, their real parts must be equal and their imaginary parts must be equal. So the above equation in sines and cosines is actually two equations, one for the real part and one for the imaginary part:
(48) cos(A+B) = cos A cos B − sin A sin B
sin(A+B) = sin A cos B + cos A sin B
In just a few short steps, the formulas for cos(A+B) and sin(A+B) flow right from equation 47, Euler’s equation for eix. No more need to memorize which one has the minus sign and how all the sines and cosines fit on the right-hand side: all you have to do is a couple of substitutions and a multiply.
Example: What’s the exact value of cos 75° or cos(5π/12)? Solution: 75° = 45°+30° (5π/12 = π/4+π/6). Using equation 48,
cos 75° = cos(45°+30°)
cos 75° = cos 45° cos 30° − sin 45° sin 30°
cos 75° = [(√2)/2]×[(√3)/2] − [(√2)/2]×[1/2]
cos 75° = (√6)/4 − (√2)/4
What about the formulas for the differences of angles? You can write them down at once from equation 48 by substituting −B for B and using equation 22. Or, if you prefer, you can get them by substituting x = A−B in equation 47 above. Either way, you get
(49) cos(A−B) = cos A cos B + sin A sin B
sin(A−B) = sin A cos B − cos A sin B
I personally find the algebraic reasoning given above very easy to follow, though you do have to remember Euler’s formula.
If you prefer geometric derivations of sin(A±B) and cos(A±B), you’ll find a beautiful set by Len and Deborah Smiley at <http://www.maa.org/pubs/mm_supplements/smiley/trigproofs.html> or <http://math.uaa.alaska.edu/~smiley/trigproofs.html>, worth the wait for the page to load. (Phil Kenny drew my attention to this page’s original version and to the link at the University of Alaska.) Eric’s Treasure Trove of Mathematics has smaller versions of the pictures at <http://mathworld.wolfram.com/TrigonometricAdditionFormulas.html>.
The fallback position is the standard proof: draw a diagram and use the distance formula or Pythagorean Theorem to prove the formula for cos(A−B).
Sometimes (not very often) you have to deal with the tangent of the sum or difference of two angles. I have only a vague idea of the formula, but it’s easy enough to work out “on the fly”:
tan(A+B) = sin(A+B) / cos(A+B)
tan(A+B) = (sin A cos B + cos A sin B) / (cos A cos B − sin A sin B)
What a mess! There’s no way to factor that and remove common terms—or is there? Suppose you start with a vague idea that you’d like to know tan(A+B) in terms of tan A and tan B rather than all those sines and cosines. The numerator and denominator contain sines and cosines, so if you divide by cosines you’d expect to end up with sines or perhaps sines over cosines. But sine/cosine is tangent, so this seems like a promising line of attack. Since you’ve got cosines of angles A and B to contend with, try dividing the numerator and denominator of the fraction by cos A cos B:
tan(A+B) = (sin A cos B + cos A sin B) / (cos A cos B − sin A sin B)
tan(A+B) = [sin A/cos A + sin B/cos B] / [1 − sin Asin B/cos Acos B]
Hmmm, looks like this is the right track. Simplify it using the definition of tan x (see equation 3), and you have
(50) tan(A+B) = (tan A + tan B) / (1 − tan A tan B)
Now if you replace B with −B, you have the formula for tan(A−B). (Take a minute to review why tan(−x) = −tan(x).)
(51) tan(A−B) = (tan A − tan B) / (1 + tan A tan B)
Example: What’s the exact value of tan 15° or tan(π/12)? Solution: 15° = 60°−45° (π/12 = π/3 − π/4). Therefore
tan(π/12) = tan(π/3 − π/4)
tan(π/12) = [tan(π/3) − tan(π/4)] / [1 + tan(π/3) tan(π/4)]
tan(π/12) = [(√3)− 1] / [1 + (√3)×1]
tan(π/12) = [(√3)− 1] / [(√3) + 1]
If you like, you can rationalize the denominator:
tan(π/12) = [(√3)− 1]² / [(√3) + 1]×[(√3) − 1]
tan(π/12) = [3 − 2(√3) + 1] / [3 − 1]
tan(π/12) = [4 − 2(√3)] / 2
tan(π/12) = 2 − √3
Advice to the reader: The formulas in this section aren’t really very useful in trigonometry itself, but are used in calculus. You may wish to skip them, especially on a first reading.
Sometimes you need to simplify an expression like cos 3x cos 5x. Of course it’s not equal to cos(15x²), but can it be simplified at all? The answer is yes, and in fact you need this technique for calculus work. There are four formulas that can be used to break up a product of sines or cosines.
These product-to-sum formulas come from the formulas in equation 48 and equation 49 for sine and cosine of A±B. First let’s develop one of these formulas, and then we’ll look at an application before developing the others.
Take the two formulas for cos(A±B) and add them:
cos(A−B) = cos A cos B + sin A sin B
cos(A+B) = cos A cos B − sin A sin B
cos(A−B) + cos(A+B) = 2 cos A cos B
½ [cos(A−B) + cos(A+B)] = cos A cos B
Example: Suppose you need to graph the function
f(x) = cos 2x cos 3x,
or perhaps you need to find its integral. Both of these are rather hard to do with the function as it stands. But you can use the product-to-sum formula, with A = 2x and B = 3x, to rewrite the function as a sum:
f(x) = cos 2x cos 3x
f(x) = ½ [cos(2x−3x) + cos(2x+3x)]
f(x) = ½ [cos(−x) + cos 5x]
Use equation 22, cos(−x) = cos x:
f(x) = ½ [cos x + cos 5x]
f(x) = ½ cos x + ½ cos 5x
This is quite easy to integrate. And while it’s not exactly trivial to graph, it’s much easier than the original, because cos x and cos 5x are easy to graph.
The other three product-to-sum formulas come from the other three ways to add or subtract the formulas in equation 48 and equation 49. If you subtract the two cosine formulas instead of adding:
cos(A−B) = cos A cos B + sin A sin B
−cos(A+B) = −cos A cos B + sin A sin B
you get
cos(A−B) − cos(A+B) = 2 sin A sin B
½ [cos(A−B) − cos(A+B)] = sin A sin B
To get the other two product-to sum formulas, add the two sine formulas from equation 48 and equation 49, or subtract them. Here are all four formulas together:
(52) cos A cos B = ½ cos(A−B) + ½ cos(A+B)
sin A sin B = ½ cos(A−B) − ½ cos(A+B)
sin A cos B = ½ sin(A+B) + ½ sin(A−B)
cos A sin B = ½ sin(A+B) − ½ sin(A−B)
The fourth one of those formulas really isn’t needed, because you can always evaluate cos p sin q as sin q cos p. But it’s traditional to present all four formulas.
There are also formulas that combine a sum or difference into a product. Heon Joon Choi, a physics student from Cornell, has kindly told me of an application: “superposing two waves and trying to figure out the nodes is much easier if they are multiplied, rather than added.” This makes sense: solving most equations is easier once you’ve factored them. The sum-to-product formulas are also used to prove the Law of Tangents, though that itself is no longer used in solving triangles.
Here’s how to get the sum-to-product formulas. First make these definitions:
A = ½(u+v), and B = ½(u−v)
Then you can see that
A+B = u, and A−B = v
Now make those substitutions in all four formulas of equation 52, and after simplifying you will have the sum-to-product formulas:
(53) cos u + cos v = 2 cos(½(u+v)) cos(½(u−v))
cos u − cos v = −2 sin(½(u+v)) sin(½(u−v))
sin u + sin v = 2 sin(½(u+v)) cos(½(u−v))
sin u − sin v = 2 sin(½(u−v)) cos(½(u+v))
Trig without Tears Part 8:
Summary: Very often you can simplify your work by expanding something like sin(2A) or cos(½A) into functions of plain A. Sometimes it works the other way and a complicated expression becomes simpler if you see it as a function of half an angle or twice an angle. The formulas seem intimidating, but they’re really just variations on equation 48 and equation 50.
With equation 48, you can find sin(A+B). What happens if you set B=A?
sin(A+A) = sin A cos A + cos A sin A
But A+A is just 2A, and the two terms on the right-hand side are equal. Therefore:
sin(2A) = 2 sin A cos A
The cosine formula is just as easy:
cos(A+A) = cos A cos A − sin A sin A
cos(2A) = cos²A − sin²A
Though this is valid, it’s not completely satisfying. It would be nice to have a formula for cos(2A) in terms of just a sine or just a cosine. Fortunately, we can use sin²x + cos²x = 1 to eliminate either the sine or the cosine from that formula:
cos(2A) = cos²A − sin²A = cos²A − (1 − cos²A) = 2 cos²A − 1
cos(2A) = cos²A − sin²A = (1 − sin²A) − sin²A = 1 − 2 sin²A
On different occasions you’ll have occasion to use all three forms of the formula for cos(2A). Don’t worry too much about where the minus signs and 1s go; just remember that you can always transform any of them into the others by using good old sin²x + cos²x = 1.
(59) sin(2A) = 2 sin A cos A
cos(2A) = cos²A − sin²A = 2 cos²A − 1 = 1 − 2 sin²A
There’s a very cool second proof of these formulas, using Sawyer’s marvelous idea. Also, there’s an easy way to find functions of higher multiples: 3A, 4A, and so on.
To get the formula for tan(2A), you can either start with equation 50 and put B = A to get tan(A+A), or use equation 59 for sin(2A) / cos(2A) and divide both parts of the fraction by cos²A. Either way, you get
(60) tan(2A) = 2 tan A / (1 − tan²A)
What about the formulas for sine, cosine, and tangent of half an angle? Since A = (2A)/2, you might expect the double-angle formulas equation 59 and equation 60 to be some use. And indeed they are, though you have to pick carefully.
For instance, sin(2A) isn’t much help. Put A = B/2 and you have
sin B = 2 sin(B/2) cos(B/2)
That’s true enough, but there’s no easy way to solve for sin(B/2) or cos(B/2).
There’s much more help in equation 59 for cos(2A). Put 2A = B or A = B/2 and you get
cos B = cos²(B/2) − sin²(B/2) = 2 cos²(B/2) − 1 = 1 − 2 sin²(B/2)
Use just the first and last parts of that:
cos B = 1 − 2 sin²(B/2)
Rearrange a bit:
sin²(B/2) = (1 − cos B) / 2
and take the square root
sin(B/2) = ± sqrt([1 − cos B] / 2)
You need the plus or minus sign because sin(B/2) may be positive or negative, depending on B. For any given B (or B/2) there will be only one correct sign, which you already know from the diagram that we explored back in Functions of Any Angle.
Example: If B = 280°, then B/2 = 140°, and you know that sin 140° is positive because the angle is in Quadrant II (above the axis).
To find cos(B/2), start with a different piece of the cos(2A) formula from equation 59:
cos(2A) = 2 cos²A − 1
As before, put A = B/2:
cos B = 2 cos²(B/2) − 1
Rearrange and solve for cos(B/2):
cos²(B/2) = [1 + cos B]/2
cos(B/2) = ± sqrt([1 + cos B] / 2)
You have to pick the correct sign for cos(B/2) depending on the value of B/2, just as you did with sin(B/2). But of course the sign of the sine is not always the sign of the cosine.
(61) sin(B/2) = ± sqrt([1 − cos B] / 2)
cos(B/2) = ± sqrt([1 + cos B] / 2)
You can find tan(B/2) in the usual way, dividing sine by cosine from equation 61:
tan(B/2) = sin(B/2)/cos(B/2) = ± sqrt([1 − cos B]/[1 + cos B])
In the sine and cosine formulas we couldn’t avoid the square roots, but in this tangent formula we can. Multiply top and bottom by sqrt(1+cos B):
tan(B/2) = sqrt([1 − cos B] / [1 + cos B])
tan(B/2) = sqrt([1−cos B][1+cos B] / [1+cos B]²)
tan(B/2) = [sqrt(1−cos²B)] / [1+cos B]
Then use equation 38, your old friend: sin²x + cos²x = 1;
tan(B/2) = sqrt(sin²B) / (1+cos B) = sin B / (1+cos B)
If you had multiplied top and bottom by sqrt(1−cos B) instead of sqrt(1+cos B), you would get
tan(B/2) = sin(B/2)/cos(B/2) = ± sqrt([1−cos B]/[1+cos B])
tan(B/2) = sqrt([1−cos B] / [1+cos B])
tan(B/2) = sqrt([1−cos B]² / [1+cos B][1−cos B])
tan(B/2) = (1−cos B) / sqrt(1−cos²B)
tan(B/2) = (1−cos B) / sqrt(sin²B) = (1−cos B) / sin B
The half-angle tangent formulas can be summarized like this:
(62) tan(B/2) = (1 − cos B) / sin B = sin B / (1 + cos B)
You may wonder what happened to the plus or minus sign in tan(B/2). Fortuitously, it drops out. Since cos B is always between −1 and +1, (1 − cos B) and (1 + cos B) are both positive for any B. And the sine of any angle always has the same sign as the tangent of the corresponding half-angle.
Don’t take my word for that last statement, please. There are only four possibilities, and they’re easy enough to work out in a table. (Review interval notation if you need to.)
| B/2 | Q I, (0°;90°) | Q II, (90°;180°) | Q III, (180°;270°) | Q IV, (270°;360°) |
|---|---|---|---|---|
| tan(B/2) | + | − | + | − |
| B | (0°;180°), Q I or II | (180°;360°), Q III or IV | (360°;540°), Q I or II | (540°;720°), Q III or IV |
| sin B | + | − | + | − |
| 1−cos B | + | + | + | + |
Of course, you can ignore the whole matter of the sign of the sine and just assign the proper sign when you do the computation.
Another question you may have about equation 62: what happens if cos B = −1, so that (1 + cos B) = 0? Don’t we have division by zero then? Well, take a little closer look at those circumstances. The angles B for which cos B = −1 are ±180°, ±540°, and so on. But in this case the half angles B/2 are ±90°, ±270°, and so on: angles for which the tangent is not defined anyway. So the problem of division by zero never arises.
And in the other formula, sin B = 0 is not a problem. Excluding the cases where cos B = −1, this corresponds to B = 0°, ±360°, ±720°, etc. But the half angles B/2 are 0°, ±180°, ±360°, and so on. For all of them, tan(B/2) = 0, as you can verify from the second half of formula equation 62.
Trig without Tears Part 9:
Summary: The inverse trig functions (also called arcfunctions) are similar to any other inverse functions: they go from the function value back to the angle (or number). Their ranges are restricted, by definition, because an inverse function must not give multiple answers. Once you understand the inverse functions, you can simplify composite functions like sin(Arctan x).
Sometimes you have a sine or cosine or tangent and need to find the associated angle. For instance, this happens whenever you solve a triangle. When you have a sine function value and find the corresponding angle, we say you are finding the arcsine or inverse sine of that value, and similarly for the other functions.
Different books use different notation: sin with a superscript −1, or arcsin. I prefer the “arc” forms because the superscript −1 looks too much like an exponent.
You may be less intimidated by the arcfunctions if you pronounce them as the angle whose. For instance, arccos 0.6 becomes “the angle whose cosine is 0.6”.
What is arcsin(0.5)? You probably recognize that 0.5 is ½, and it must be a sine of one of the special angles. In fact, equation 15 tells you that sin(30°) = ½. So you can say that arcsin(0.5) = 30° or π/6.
But wait, there’s more! You know from equation 22 that
sin(30°) = sin(150°) = sin(390°) = sin(−210°)
and so on, and therefore they all equal ½. In fact,
sin(30° + 360°k) = sin(150° + 360°k) = 0.5 for all integer k
So which of this infinite number of values is the arcsine?
To make an arcsine function, we have to restrict the range (the possible answers given by each function) so that each number has at most one arcsine. (Why do I say “at most one” and not “one and only one”? The sine of any angle is between −1 and +1 inclusive; therefore only those numbers have arcsines. There is o angle whose sine is 2, so 2 has no arcsines and arcsin 2 is a meaningless noise.)
The arcsine is defined so that its range is the interval [−π/2;+π/2], which is the same as [−90°;+90°]. Some books use a capital letter for the function Arcsin distinguish it from the multi-valued relation arcsin. So we could say
arcsin(0.5) = π/6+2kπ or 5π/6+2kπ for all integer k
but
Arcsin(0.5) = π/6
Why the particular range −π/2 to +π/2? To start with, it seems tidy that any arcfunction of a positive number should fall in Quadrant I, [0;+π/2]. So the only real question is arcfunctions of negative numbers. If we prefer the numerically smallest values for the arcsine function, then Arcsin(−0.5) = −30° = −π/6 fits that rule, and a negative number’s arcsine (and arctangent, too) will be in Quadrant IV, [−π/2;0].
What about the arccosine? The cosine is positive in both Quadrant I and Quadrant IV, so the arccosine of a negative number must fall in Quadrant II or Quadrant III. Thomas (Calculus and Analytic Geometry, 4th edition) resolves this in a neat way. Remember from equation 2 that
cos A = sin(π/2 − A)
It makes a nice symmetry to write
Arccos x = π/2 − Arcsin x
And that is how Thomas defines the inverse cosine function. Since the range of Arcsin is the closed interval [−π/2;+π/2], the range of Arccos is π/2 minus that, or [0;π].
You can remember the range of the arctangent function this way: the only two choices that make sense are 0 to π and −π/2 to +π/2. But tan(π/2) is undefined, so if you picked [0;π] for the range of Arctan it would have an ugly hole in it at π/2. Therefore the range is defined to be (−π/2;+π/2), an open interval but with no points missing inside.
Once the range for Arctan is defined, there’s really only one sensible way to define Arccot:
cot x = tan(π/2 − x) ⇒ Arccot x = π/2 − Arctan x
which gives the single open interval (0;π) as the range.
Thomas defines the arcsecant and arccosecant functions using the reciprocal relationships from equation 5:
sec x = 1/(cos x) ⇒ Arcsec x = Arccos(1/x)
csc x = 1/(sin x) ⇒ Arccsc x = Arcsin(1/x)
This means that Arcsec and Arccsc have the same ranges as Arccos and Arcsin, respectively.
Here are the domains (inputs) and ranges (outputs) of all six inverse trig functions:
| function | derived from | domain | range |
|---|---|---|---|
| Arcsin | inverse of sine function | [−1; +1] | [−π/2; +π/2] |
| Arccos | Arccos x = π/2 − Arcsin x | [−1; +1] | [0; π] |
| Arctan | inverse of tangent function | all reals | (−π/2; +π/2) |
| Arccot | Arccot x = π/2 − Arctan x | all reals | (0; π) |
| Arcsec | Arcsec x = Arccos(1/x) | (−∞; −1], [1; ∞) | [0; π] |
| Arccsc | Arccsc x = Arcsin(1/x) | (−∞; −1], [1; ∞) | [−π/2; +π/2] |
Remember that the relations are many-valued, not limited to the above ranges of the functions. If you see the capital A in the function name, you know you’re talking about the function; otherwise you have to depend on context.
Advice to the reader: The formulas in this section aren’t really very useful in trigonometry itself, but are used in integral calculus and some physics or engineering courses. You may wish to skip them, especially on a first reading.
Sometimes you have to evaluate expressions like
cos(Arctan x)
That looks scary, but actually it’s a piece of cake. You can simplify any trig function of any inverse function in two easy steps, using this method:
Think of the inner arcfunction as an angle. Draw a right triangle and label that angle and the two relevant sides.
Use the Pythagorean Theorem to find the third side of the triangle, then write down the value of the outer function according to its definition.
(If the answer contains variables raised to odd powers, you may need to add some absolute value signs. See Example 3.)
It may be helpful to read the expression out in words: “the cosine of Arctan x.” Doesn’t help much? Well, remember what Arctan x is. It’s the (principal) angle whose tangent is x. So what you have to find reads as “the cosine of the angle whose tangent is x.” And that suggests your plan of attack: first identify that angle, then find its cosine.
Let’s give a name to that “angle whose”. Call it A:
A = Arctan x
from which you know that
tan A = x
Now all you have to do is find cos A, and that’s easy
if you draw a little picture.
Start by drawing a right triangle, and mark one acute angle as A.
Using the definition of A, write down the lengths of two sides of the triangle. Since tan A = x, and the definition of tangent is opposite side over adjacent side, the simplest choice is to label the opposite side x and the adjacent side 1. Then, by definition, tan A = x/1 = x, which is how we needed to set up angle A.
The next step is to find the third side. Here you know the two legs,
so you use the theorem of Pythagoras to find the hypotenuse,
sqrt(1+x²). (For some problems, you’ll know one leg and the
hypotenuse, and you’ll use the theorem to find the other leg.)
Once you have all three sides’ lengths, you can write down the value of any function of A. In this case you need cos A, which is adjacent side over hypotenuse:
cos A = 1/sqrt(1+x²)
But cos A = cos(Arctan x). Therefore
cos(Arctan x) = 1/sqrt(1+x²)
and there’s your answer.
Read this as “the cosine of the angle A whose sine is x”. Draw your triangle, and label angle A. (Please take a minute and make the drawing.) You know from equation 1 that
sin A = x = opposite/hypotenuse
and therefore you label the opposite side x and the hypotenuse 1.
Next, solve for the third side, which is sqrt(1−x²), and write that down. Now you need cos A, which is the adjacent side over the hypotenuse, which is sqrt(1−x²)/1. Answer:
cos(Arcsin x) = sqrt(1−x²)
There you go: quick and painless.
This looks similar to Example 1, but as you’ll see there’s an additional wrinkle. (Thanks to Brian Scott, who raised the issue, albeit inadvertently, in his article “Re: Expression” on 10 Dec 2000 in alt.algebra.help.)
Begin in the regular way by drawing your triangle.
Since A =
Arctan(1/x), or tan A = 1/x, you
make 1 the length of the opposite side and x the length of the
adjacent side. The hypotenuse is then sqrt(1+x²).
Now you can write down cos A, which is adjacent over hypotenuse:
cos A = x / sqrt(1+x²)
cos(Arctan 1/x) = x / sqrt(1+x²)
But this example has a problem that does not occur in the earlier examples.
Suppose x is negative, say −√3. Then Arctan(−1/√3) = −π/6, and cos(−π/6) = +(√3)/2. But the answer in the previous paragraph, x/sqrt(1+x²), yields −√3/√(1+3) = −(√3)/2, which has the wrong sign.
What went wrong? The trouble is that Arctan yields values in (−π/2;+π/2), which is Quadrants IV and I. But the cosine is always positive on that interval. Therefore cos(Arctan x) always yields a positive result. Remember also from equation 22 that cos(−A) = cos A. To ensure this, use the absolute value sign, and the true final answer is
cos(Arctan(1/x)) = | x | / sqrt(1+x²)
Why doesn’t every example have this problem? The earlier examples involved only the square of a variable, which is naturally nonnegative. Only here, where we have an odd power, does it matter.
Summary: When your answer contains an odd power (1, 3, 5, etc.) of a variable, you must add a third step to the process: carefully examine the signs and adjust your answer so that it has the correct sign for both positive and negative values of the variable.
Advice to the reader: The formulas in this section are for the really hard-core trig fan. They aren’t really very useful in trigonometry itself, but are used in integral calculus and some physics or engineering courses. You may wish to skip them, especially on a first reading.
You may be wondering about the inside-out versions, taking the arcfunction of a function. Some of these expressions can be solved algebraically, on a restricted domain at least, but some cannot. (I am grateful to David Cantrell for help with analysis of these problems in general and Example 6 in particular.)
We can say at once that there will be no pure algebraic equivalent to an arcfunction of a trig function. This means there will be no nice neat procedure as there was for functions of arcfunctions
Why? The six trig functions are all periodic, and therefore any function of any of them must also be periodic. But no algebraic functions are periodic, except trivial ones like f(x) = 2, and therefore no function of a trig function can be represented by purely algebraic operations. As we will see, some can be represented if we add non-algebraic functions like mod and floor.
This is the angle whose cosine is sin u. To come up with a simpler form, set x equal to the desired expression, and solve the equation by taking cosine of both sides:
x = Arccos(sin u)
cos x = sin u
This could be solved if we could somehow transform it to sin(something) = sin(u) or cos(x) = cos(something else). In fact, we can use equation 2 to do that. It tells us that
sin u = cos(π/2 − u)
and combining that with the above we have
cos x = cos(π/2 − u)
Now if x is in Quadrant I, the interval [0;π/2], then u will be in Quadrant I also, and we can write
x = π/2−u
and therefore
Arccos(sin u) = π/2−u for u in Quadrant I
But this solution does not work for all quadrants. For instance, try a number from Quadrant II:
Arccos(sin(5π/6)) = Arccos(½) = π/6
but
π/2 − 5π/6 = −π/3
Obviously π/2−u isn’t a general solution for Arccos(sin u). Try graphing Arccos(sin x) and π/2−x and you’ll see the problem: one is a sawtooth and the other is a straight line.
Sparing you the gory details, π/2−u is right only in Quadrants IV and I. We have to “decorate” it rather a lot to make it match Arccos(sin u) in the other quadrants, and also to account for the repetition of values every 2π. The first modification is not too hard: On the interval [−π/2;+3π/2], the absolute-value expression |π/2−u| matches the sawtooth graph of Arccos(sin u).
The repetition every 2π is harder to reflect, but this manages it:
Arccos(sin u) = | π/2 − u + 2π*floor[(u+π/2)/2π] |
where “floor” means the greatest integer less than or equal to. Messy, eh? (Note also that “floor” is not an algebraic function.)
It could be made a bit shorter with mod (which is also not algebraic):
Arccos(sin u) = | π − mod(u+π/2,2π) |
where mod(a,b) is the nonnegative remainder when a is divided by b.
This one, the angle whose secant is cos u, has a very odd solution. Try the solution method from Example 4 and you get
x = Arcsec(cos u)
sec x = cos u
But sec x = 1/cos x, and therefore
1/( cos x ) = cos u
Now think about that equation. The cosine’s values are all between −1 and +1. So the only way one cosine can be the reciprocal of another is if they’re both equal to 1 or both equal to −1; no other solutions exist.
First case: If cos u = 1 then u is an even multiple of π. But Arcsec 1 = 0, and therefore
Arcsec(cos u) = 0 when u = 2kπ
Second case: If cos u = −1 then u is an odd multiple of π. But Arcsec(−1) = π, and therefore
Arcsec(cos u) = π when u = (2k+1)π
If u is not a multiple of π, cos u will be less than 1 and greater than −1. The Arcsec function is not defined for such values, and therefore
Arcsec(cos u) does not exist when u is not a multiple of π
The graph of Arcsec(cos u) is rather curious: single points at the ends of an infinite sawtooth: ..., (−3π,π), (−2π,0), (−π,π), (0,0), (π,π), (2π,0), (3π,π), ...
Proceeding in the regular way, we have
x = Arctan(sin u)
tan x = sin u
The most likely approach is the one from Example 4: try to transform the above into tan(x) = tan(something) or sin(something else) = sin(u).
If there is any trig identity or combination that can be used to do that, it is unknown to me. I suspect strongly that Arctan(sin u) can’t be converted to an algebraic expression, even with the use of mod or floor, but I can’t prove it.
Trig without Tears:
Dad sighed. “Kip, do you think that table was brought down from on high by an archangel?”
Robert A. Heinlein, in Have Space Suit—Will Travel (1958)
It’s not just that there are so many trig identities; they seem so arbitrary. Of course they’re not really arbitrary, since all can be proved; but when you try to memorize all of them they seem like a jumble of symbols where the right ones aren’t more obviously right than the wrong ones. For example, is it sec²A = 1 + tan²A or tan²A = 1 + sec²A ? I doubt you know off hand which is right; I certainly don’t remember. Who can remember a dozen or more like that, and remember all of them accurately?
Too many teachers expect (or allow) students to memorize the trig identities and parrot them on demand, much like a series of Bible verses. In other words, even if they’re originally taught as a series of connected propositions, they’re remembered and used as a set of unrelated facts. And that, I think, is the problem. The trig identities were not brought down by an archangel; they were developed by mathematicians, and it’s well within your grasp to re-develop them when you need to. With effort, we can remember a few key facts about anything. But it’s much easier if we can fit them into a context, so that the identities work together as a whole.
Why bother? Well, of course it will make your life easier in trig class. But you’ll also need the trig identities in later math classes, especially calculus, and in physics and engineering classes. In all of those, you’ll find the going much easier if you’re thoroughly grounded in trigonometry as a unified field of knowledge instead of a collection of unrelated facts.
This is why it’s easier to remember almost any song than an equivalent length of prose: the song gives you additional cues in the rhythm, common patterns of emphasis, and usually rhymes at the ends of lines. With prose you have only the general thought to hold it together, so that you must memorize it essentially as a series of words. With the song there are internal structures that help you, even if you’re not aware of them.
If you’re memorizing Lincoln’s Gettysburg Address, you might have trouble remembering whether he said “recall” or “remember” at a certain point; in a song, there’s no possible doubt which of those words is right because the wrong one won’t fit in the rhythm.
I’m not against all memorization. Some things have to be memorized because they’re a matter of definition. Others you may choose to memorize because you use them very often, you’re confident you can memorize them correctly, and the derivation takes more time than you’re comfortable with. Still others you may not set out to memorize, but after using them many times you find you’ve memorized them without trying to—much like a telephone number that you dial often.
I’m not against all memorization; I’m against needless memorization used as a substitute for thought. If you decide in particular cases that memory works well for you, I won’t argue. But I do hope you see the need to be able to re-derive things on the spot, in case your memory fails. Have you ever dialed a friend’s telephone number and found you couldn’t quite remember whether it was 6821 or 8621? If you can’t remember a phone number, you have to look it up in the book. My goal is to free you from having to look up trig identities in the book.
Thanks to David Dixon at [email protected] for an
illuminating exchange of notes on this topic. He made me realize that
I was sounding more more anti-memory than I intended to, and in
consequence I’ve added this note. But he may not necessarily agree
with what I say here.
I wrote these pages to show you how to make the trig identities “fit” as a coherent whole, so that you’ll have no more doubt about them than you do about the words of a song you know well. The difference is that you won’t need to do it from memory. And you’ll gain the sense of power that comes from mastering your subject instead of groping tentatively and hoping to strike the right answer by good luck.
Euler’s formula (equation 47) is easily proved by means of power series. Start with the formulas
(81) ex = SUM [ xn / n! ] = 1 + x + x2/2! + x3/3! + ...
cos x = SUM [ (−1)n x2n / (2n)! ] = 1 − x2/2! + x4/4! − x6/6! + ...
sin x = SUM [ (−1)n x2n+1 / (2n+1)! ] = x − x3/3! + x5/5! − x7/7! + ...
(These are how the function values are actually calculated, by the way. If you want to know the value of e2, you just substitute 2 for x in the formula and compute until the additional terms fall within your desired accuracy.)
Now we have to find the value of eix, where i = √(−1). Use the first formula to find eix, by substituting ix for x:
eix = SUM [ (ix)n / n! ]
eix = 1 + (ix) + (ix)2/2! + (ix)3/3! + (ix)4/4! + (ix)5/5! + (ix)6/6! + (ix)7/7! + ...
Simplify the powers of i, using i² = −1:
eix = 1 + ix − x2/2! − ix3/3! + x4/4! + ix5/5! − x6/6! − ix7/7! + ...
Finally, group the real and imaginary terms separately:
eix = [1 − x2/2! + x4/4! − x6/6! + ...] + i[x − x3/3! + x5/5! − x7/7! + ...]
Those should look familiar. If you refer back to the power series at the start of this section you’ll see that the first group of terms is just cos x and the second group is just sin x. So you have
eix = cos x + i sin x
which is Euler’s formula, as advertised!
You may wonder where the series for cos x, sin x, and ex come from. The answer is that they are the Taylor series expansions of the functions. (You’ll probably study Taylor series in second- or third-semester calculus.)
As you may remember, complex numbers like 3+4i and
2−7i are often plotted on a complex plane. This makes
it easier to visualize adding and subtracting. The illustration at
right shows that
(3+4i) + (2−7i) = 5−3i
It turns out that for multiplying, dividing, and finding
powers and roots,
complex numbers are easier to work with in
polar form. This means that instead of thinking about the real
and imaginary components (the “−3” and “5” in
−3+5i), you think of the length and direction of the
line.
The length is easy, just the good old Pythagorean theorem in fact:
r = √(3²+5²) = √34 ≅ 5.83
(The length r is called the absolute value or modulus of the complex number.)
But what about the direction? You can see from the picture that θ about 120° or about 2 radians (measured counterclockwise from the positive real axis, which points east), but what is it exactly? Well, you know that tan θ = −5/3. If you take Arctan(−5/3) on your calculator, you get −1.03, and adding π to get into the proper quadrant gives θ = 2.11 radians. (In degrees, Arctan(−5/3) = −59.04, and adding 180 gives θ = 120.96°.) You can write it this way:
−3+5i ≅ 5.83 e2.11i or 5.83[cos 2.11 + i sin 2.11] or 5.83 cis 2.11 or 5.83∠120.96°
There’s a nice trick that finds the angle in the correct quadrant automatically:
θ = 2 Arctan(y/(x+r))
This builds in the adjustment to the correct quadrant, so you never have to worry about whether to add or subtract 180° or π. If you test 2 Arctan(5/(−3+√34)) on your calculator, you get 2.11 radians or 120.96° as before.
This formula is particularly handy when you’re writing a program or spreadsheet formula to find θ. It flows directly from the formulas for tangent of half an angle, where the functions are expressed in terms of x, y, and r. A July 2003 article by Rob Johnson, archived here, inspired me to add this section.
One of the applications of Euler’s formula (equation 47) is finding any power or root of any complex number. (Sawyer doesn’t do this, or at least not in the same book.) It’s not hard to develop the formula, using just Euler’s formula and the laws of exponents.
Start by writing down Euler’s formula, multiplied left and right by a scale factor r:
r (cos x + i sin x) = r eix
Next raise both sides to the Nth power:
[r (cos x + i sin x)]N = [r eix]N
Let’s leave the left-hand side alone for a while, and simplify the right-hand side. First, by the laws of exponents
[r eix]N = rN eiNx
Apply Euler’s formula to the right-hand side and you have
[r eix]N = rN (cos Nx + i sin Nx)
Connect that right-hand side to the original left-hand side and you have DeMoivre’s theorem:
(82) [r(cos x + i sin x)]N = rN (cos Nx + i sin Nx)
This tells you that if you put a number into polar form, you can find any power or root of it. (Remember that the nth root of a number is the same as the 1/n power.)
For instance, you know the square roots of −1 are i and −i, but what’s the square root of i? You can use DeMoivre’s theorem to find it.
First, put i into eix form (polar form), using the above technique. i = 0+1i. cos x = 0 and sin x = 1 when x = 90° or π/2. Therefore
i = cos(π/2) + i sin(π/2)
√i = i½ = [cos(π/2) + i sin(π/2)]½
And by equation 82
√i = cos(½×π/2) + i sin(½×π/2)
√i = cos(π/4) + i sin(π/4)
√i = 1/√2 + i×1/√2
√i = (1+i)/√2
The other square root is minus that, as usual.
You can find any root of any complex number in a similar way, but usually with one preliminary step.
For instance, suppose you want the cube roots of 3+4i. The first step is to put that number into polar form. The absolute value is sqrt(3²+4²) = 5, and you use the Arctan technique given above to find the angle θ = 2 Arctan(4/(3+5)) = 2 Arctan(½) ≅ 53.13° or 0.9273 radians. The polar form is
3+4i ≅ 5(cos 53.13° + i sin 53.13°)
To take a cube root of that, use equation 82 with N = 1/3:
cube root of (3+4i) = 51/3 [cos(53.13°/3) + i sin(53.13°/3)]
The cube root of 5 is about 1.71. Dividing the angle θ by 3, 53.13°/3 ≅ 17.71°; the cosine and sine of that are about 0.95 and 0.30. Therefore
cube root of (3+4i) ≅ 1.71 × (cos 17.71° + i sin 17.71°)
cube root of (3+4i) ≅ 1.63 + 0.52i
You may have noticed that I talked about “the cube roots [plural] of 3+4i” and what we found was “a cube root”. Even with the square root of i, I waved my hand and said that the “other” square root was minus the first one, “as usual”.
You already know that every positive real has two square roots. In fact, every complex number has n nth roots.
How can you find them? It depends on the periodic nature of the cosine and sine functions. To start, look back at Euler’s formula,
eix = cos x + i sin x
What happens if you add 2π or 360° to x? You have
ei(x+2π) = cos(x+2π) + i sin(x+2π)
But taking sine or cosine of 2π plus an angle is exactly the same as taking sine or cosine of the original angle. So the right-hand side is equal to cos x + i sin x, which is equal to eix. Therefore
ei(x+2π) = eix
In fact, you can keep adding 2π or 360° to x as long as you like, and never change the value of the result. Symbolically,
ei(x+2πk) = eix for all integer k
When you take an nth root, you simply use that identity.
Getting back to the cube roots of 3+4i, recall that that number is the same as 5eiθ, where θ is about 53.13° or 0.9273 radians. The three cube roots of eiθ are
eiθ/3, ei(θ+2π)/3, and ei(θ+4π)/3 for k = 0,1,2
or
eiθ/3, ei(θ/3+2π/3), and ei(θ/3+4π/3) for k = 0,1,2
Compute those as cos x + i sin x in the usual way, and then multiply by the (principal) cube root of 5. I get these three roots:
cube roots of 3+4i ≅ 1.63+0.52i, −1.26+1.15i, −0.36−1.67i
More generally, we can extend DeMoivre’s theorem (equation 82) to show the N Nth roots of any complex number:
(83) [r (cos x + i sin x](1/N) = r(1/N) [cos(x/N+2πk/N) + i sin(x/N+2πk/N)] for k = 0,1,2,...,N−1
Another application flows from a famous special case of Euler’s formula. Substitute x = π or 180° in equation 47. Since sin 180° = 0, the imaginary term drops out. And cos 180° = −1, so you have the famous formula
(84) −1 = eiπ
It’s interesting to take the natural log of both sides:
ln(−1) = ln(eiπ)
which gives
ln(−1) = iπ
It’s easy enough to find the logarithm of any other negative number. Since
ln(ab) = ln a + ln b
then for all real a you have
ln(−a) = ln[a × −1] = ln a + ln(−1) = ln a + iπ
(85) ln(−a) = ln a + iπ
I don’t honestly know whether all of this has any practical application. But if you’ve ever wondered about the logarithm of a negative number, now you know.
I can’t resist pointing out another cool thing about Sawyer’s marvelous idea: you can also use it to prove the double-angle formulas equation 59 directly. From Euler’s formula for eix you can immediately obtain the formulas for cos(2A) and sin(2A) without going through the formulas for sums of angles. Here’s how.
Remember the laws of exponents: xab = (xa)b. One important special case is that x2b = (xb)². Use that with Euler’s formula (equation 47):
cos(2A) + i sin(2A) = e(2A)i
= (eiA)²
= (cos A + i sin A)²
= cos²A + 2i sinA cosA + i²sin²A
= cos²A + 2i sinA cosA − sin²A
= cos²A − sin²A + 2i sinA cosA
Since the real parts on left and right must be equal, you have the formula for cos(2A). Since the imaginary parts must be equal, you have the formula for sin(2A). That’s all there is to it.
The above technique is even more powerful for deriving formulas for functions of 3A, 4A, or any multiple of angle A. To derive the formulas for nA, expand the nth power of (cos A + i sin A), then collect real and imaginary terms.
This was inspired by a May 2009 reading of Paul Nahin’s An Imaginary Tale: The Story of √−1 (Princeton, 1998).
Just to show the method, I’ll derive the functions of 3A and 4A. You can try the brute-force approach of cos(2A+2A) and sin(2A+2A) and see how much effort my method saves.
De Moivre’s theorem (equation 82) tells us that
cos 3A + i sin 3A = (cos A + i sin A)³
Expand the right-hand side via the binomial theorem, remembering that i² = −1 and i³ = −i:
cos 3A + i sin 3A = cos³A + 3 i cos²A sin A − 3 cos A sin²A − i sin³A
Collect real and imaginary terms:
cos 3A + i sin 3A = cos³A − 3 cos A sin²A + 3 i cos²A sin A − i sin³A
cos 3A + i sin 3A = [cos A (cos²A − 3 sin²A)] + i [sin A (3 cos²A − sin²A)]
Set the real part on the left equal to the real part on the right, and similarly for the imaginary parts, and you have the formulas for the cosine and sine of 3A:
cos 3A = cos A (cos²A − 3 sin²A) and sin 3A = sin A (3 cos²A − sin²A)
The tangent formula is easy to get: just divide.
tan 3A = sin 3A / cos 3A
tan 3A = [sin A (3 cos²A − sin²A)] / [cos A (cos²A − 3 sin²A)]
tan 3A = tan A (3 cos²A − sin²A) / (cos²A − 3 sin²A)
Divide top and bottom by cos²A to simplify:
tan 3A = tan A (3 − tan²A) / (1 − 3 tan²A)
(86) cos 3A = cos A (cos²A − 3 sin²A)
sin 3A = sin A (3 cos²A − sin²A)
tan 3A = tan A (3 − tan²A) / (1 − 3 tan²A)
It’s no more work to find the functions of 4A. I’ll just run through the steps without commentary. First, cosine and sine of 4A:
cos 4A + i sin 4A = (cos A + i sin A)4
cos 4A + i sin 4A = cos4A + 4 i cos³A sin A − 6 cos²A sin²A − 4 i cos A sin³A + sin4A
cos 4A + i sin 4A = cos4A − 6 cos²A sin²A + sin4A + 4 i cos³A sin A − 4 i cos A sin³A
cos 4A = cos4A − 6 cos²A sin²A + sin4A and sin 4A = 4 cos³A sin A − 4 cos A sin³A
cos 4A = (cos²A − sin²A)² − 4 sin²A cos²A and sin 4A = 4 sin A cos A (cos²A − sin²A)
Now the tangent formula:
tan 4A = sin 4A / cos 4A
tan 4A = [4 sin A cos A (cos²A − sin²A)] / [(cos²A − sin²A)² − 4 sin²A cos²A]
Divide top and bottom by cos4A:
tan 4A = 4 tan A (1 − tan²A) / [(1 − tan²A)² − 4 tan²A]
(87) cos 4A = (cos²A − sin²A)² − 4 sin²A cos²A
sin 4A = 4 sin A cos A (cos²A − sin²A)
tan 4A = 4 tan A (1 − tan²A) / [(1 − tan²A)² − 4 tan²A]
I have to recommend a terrific little book, How To Solve It by G. Polya. Most teachers aren’t very good at teaching you how to solve problems and do proofs. They show you how they do them, and expect you to pick up their techniques by a sort of osmosis. But most of them aren’t very good at explaining the thought process that goes into doing a geometrical proof, or solving a dreaded “story problem”.
Polya’s book does a great job of teaching you how to solve problems. He shows you the kinds of questions you should ask yourself when you see a problem. In other words, he teaches you how to get yourself over the hum, past the floundering that most people do when they see an unfamiliar problem. And he does it with lots of examples, so that you can develop confidence in your techniques and compare your methods with his. The techniques I’ve mentioned above are just three out of the many in his book.
There’s even a handy checklist of questions you can ask yourself whenever you’re stuck on a problem.
How To Solve It was first published in 1945, and it’s periodically in and out of print. If you can’t get it from your bookstore, go to the library and borrow a copy. You won’t be sorry.
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