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10-Minute Trig

revised 28 Mar 2006

Copyright © 2002–2013 by Stan Brown, Oak Road Systems

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

triangle A=45, B=45, C=90, c=1 equilateral triangle, side=1, bisected vertically; bisector=a, angle A=60
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)

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