revised 28 Mar 2006
Copyright © 2002–2012 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.
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.
[MODE] to check calculator mode (radian
|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
The definitions based on an acute angle in a right triangle extend to trig functions of any angle:
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.
|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.|
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)