10-Minute Trig

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

C. Trig Functions of Any Angle

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° = 1/1 = 1 and sin π = 0/−1 = 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

E. Inverse Functions

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 A. Then you must find cos A. Use Pythagoras to find the third side, √x²+1, then read off function value: cos A = 1 / √x²+1.