Mathematics · Ch 3 — Trigonometric Functions

Summary

Angles. An angle is a signed rotation (anticlockwise positive,

clockwise negative) of a ray about its endpoint; two angles with the same terminal side,

differing by a whole number of revolutions, are coterminal.

Degree and radian measure. $\pi$ radians $=180^\circ$ ; degrees $\to$ radians: multiply by

$\pi/180$ ; radians $\to$ degrees: multiply by $180/\pi$ . Arc length: $s=r\theta$ ( $\theta$ in

radians).

Unit-circle definition. For any real $\theta$ , with $P(\cos\theta,\sin\theta)$ the point

where the terminal side of $\theta$ meets the unit circle: $\cos\theta=x$ , $\sin\theta=y$ ,

and $\tan\theta,\cot\theta,\sec\theta,\text{cosec}\,\theta$ are the usual ratios of $x,y$ .

Fundamental identity. $\sin^2x+\cos^2x=1$ for all real $x$ ; dividing through gives

$1+\tan^2x=\sec^2x$ and $1+\cot^2x=\text{cosec}^2x$ .

Signs, domain, range, period. Signs follow the "All Sin Tan Cos" quadrant rule. $\sin,\cos$ :

domain $\mathbb{R}$ , range $[-1,1]$ , period $2\pi$ . $\tan,\cot$ : range $\mathbb{R}$ , period

$\pi$ , undefined at odd multiples of $\pi/2$ (tan) or multiples of $\pi$ (cot).

Sum and difference formulas.

\cos(x\mp y)=\cos x\cos y\pm\sin x\sin y,\qquad \sin(x\pm y)=\sin x\cos y\pm\cos x\sin y,

\tan(x\pm y)=\frac{\tan x\pm\tan y}{1\mp\tan x\tan y},\qquad \cot(x\pm y)=\frac{\cot x\cot y\mp1}{\cot y\pm\cot x}.

Sum-to-product formulas.

\sin x+\sin y=2\sin\tfrac{x+y}2\cos\tfrac{x-y}2,\qquad \sin x-\sin y=2\cos\tfrac{x+y}2\sin\tfrac{x-y}2,

\cos x+\cos y=2\cos\tfrac{x+y}2\cos\tfrac{x-y}2,\qquad \cos x-\cos y=-2\sin\tfrac{x+y}2\sin\tfrac{x-y}2.

Multiple-angle identities.

\sin2x=2\sin x\cos x,\quad \cos2x=\cos^2x-\sin^2x=2\cos^2x-1=1-2\sin^2x,\quad \tan2x=\frac{2\tan x}{1-\tan^2x},

\sin3x=3\sin x-4\sin^3x,\quad \cos3x=4\cos^3x-3\cos x,\quad \tan3x=\frac{3\tan x-\tan^3x}{1-3\tan^2x}.

General solutions. …