10-Minute Trig

revised 28 Mar 2006

Summary: Use this page to jog your memory with the basic facts of right-angle trigonometry. Please see Trig without Tears for explanations of these quick notes and many more topics.

A. Degrees and Radians

circumference of circle: 2πr
angle around circle (like clock hand): 2π radians or 360°
Therefore 2π = 360°, or π = 180°, or 1 radian = 180°/π.

Press [MODE] to check calculator mode (radian or degree).

B. Trig Ratios of Acute Angles

a right triangle ABC, with the right angle C at lower right; the sides a, b, c are opposite the angles A, B, C respectively, making c the hypotenuse of the triangle. $tan A = sin A over cos A = opp over adj = a over b$

SOHCAHTOA	sin A = cos(π/2−A) cos A = sin(π/2−A) tan A = cot(π/2−A) cot A = tan(π/2−A) sec A = csc(π/2−A) csc A = sec(π/2−A)
cot A = 1 / tan A sec A = 1 / cos A csc A = 1 / sin A

C. Trig Functions of Any Angle

The definitions based on an acute angle in a right triangle extend to trig functions of any angle:

$x,y,r approach to trig functions$
sin θ = y/r cos θ = x/r tan θ = sin θ / cos θ = y/x
cot θ = 1 / tan θ sec θ = 1 / cos θ csc θ = 1 / sin θ

Pythagorean theorem ( y² + x² = r² ) leads to sin² θ + cos² θ = 1

r is always >0, so signs of functions in any quadrant pop right out from signs of x and y in that quadrant. Do quadrant angles by reference to x y r, e.g. cos 0° = 1 and sin π = 0.

Use reference angle (acute angle between terminal side and x axis) to relate function values to values for an acute angle.

D. Trig Functions of Special Angles


c = 1 (given), a = b By Pythagoras, a = b = √2 / 2	c=1 (given), b = c/2 = ½ By Pythagoras, a = √3 / 2

	0 = 0°	π/6 = 30°	π/4 = 45°	π/3 = 60°	π/2 = 90°
sin θ	0	1/2	√2 / 2	√3 / 2	1
cos θ	1	√3 / 2	√2 / 2	1/2	0
tan θ	0	√3 / 3	1	√3	undef.

E. Inverse Functions

right triangle, acute angle=A, adjacent side=1, opposite side=x arcsin 0.65 or sin^-10.65 means the angle whose sine is 0.65. That’s not the same as 1/sin 0.65
Function ranges: −π/2 ≤ arcsin x ≤ +π/2, 0 ≤ arccos x ≤ +π, −π/2 < arctan x < +π/2

Function composition (see diagram at right):
What is e.g. cos( arctan x ) ?
Solution: arctan x is the angle whose tangent is x; call it θ. Then you must find cos θ. Use Pythagoras to find the third side, √(x²+1), then read off function value: cos θ = 1 / √(x²+1)

What’s New

28 Mar 2006: Small changes of wording for clarity; change en dashes (-) to minus signs (−), though they look the same in many fonts.
11 Jul 2003: Add radians for special angles and add special angles 0 and π/2 = 90°.
10 Jul 2003: New document, based on a summary I prepared for my precalculus class in March 2002.

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