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trigonometry

Unit circle, exact values, identities, compound & double angles, trig equations, triangle rules.

triangles · radians & unit circle · exact values · symmetry · identities · graphs & equations

Triangle trigonometry

Right-triangle ratios. \(\sin\theta=\dfrac{\text{opp}}{\text{hyp}}\), \(\cos\theta=\dfrac{\text{adj}}{\text{hyp}}\), \(\tan\theta=\dfrac{\text{opp}}{\text{adj}}\).

Sine rule. \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}=2R\).

Cosine rule. \(c^2=a^2+b^2-2ab\cos C\); \(\cos C=\dfrac{a^2+b^2-c^2}{2ab}\).

Ambiguous case (SSA). \(\sin B=b\sin A/a\); possible solutions \(B\) and \(\pi-B\) where the angle sum stays valid.

Bearings. Measured clockwise from north; angles of elevation/depression measured from the horizontal.

Radians & unit circle

Conversion. \(\theta_{\mathrm{rad}}=\theta^\circ\pi/180\), \(\theta^\circ=180\,\theta_{\mathrm{rad}}/\pi\).

Unit circle. Point at angle \(\theta\) is \((\cos\theta,\sin\theta)\); \(\tan\theta=\sin\theta/\cos\theta\). Quadrant signs: ASTC. Reference angle \(\alpha\): evaluate at \(\alpha\), attach the quadrant sign. Line through the origin: \(y=x\tan\theta\).

Exact values

Symmetry

Negative angle. \(\sin(-x)=-\sin x\), \(\cos(-x)=\cos x\), \(\tan(-x)=-\tan x\).

Supplement & antisupplement. \(\sin(\pi-x)=\sin x\), \(\cos(\pi-x)=-\cos x\), \(\tan(\pi-x)=-\tan x\); \(\sin(\pi+x)=-\sin x\), \(\cos(\pi+x)=-\cos x\), \(\tan(\pi+x)=\tan x\).

Periods. \(\sin\), \(\cos\): \(2\pi\); \(\tan\): \(\pi\).

Identities

Pythagorean. \(\sin^2x+\cos^2x=1\); \(1+\tan^2x=\sec^2x\); \(1+\cot^2x=\cosec^2x\).

Reciprocal. \(\sec x=1/\cos x\), \(\cosec x=1/\sin x\), \(\cot x=1/\tan x=\cos x/\sin x\).

Compound angles. \(\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B\); \(\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B\); \(\tan(A\pm B)=\dfrac{\tan A\pm\tan B}{1\mp\tan A\tan B}\).

Double angles. \(\sin 2x=2\sin x\cos x\); \(\cos 2x=\cos^2x-\sin^2x=1-2\sin^2x=2\cos^2x-1\); \(\tan 2x=\dfrac{2\tan x}{1-\tan^2x}\).

Graphs & equations

Sinusoids. \(y=a\sin(b(x+c))+d\): amplitude \(|a|\), period \(2\pi/|b|\), phase shift \(-c\), midline \(y=d\); cosine the same.

Tangent. \(y=a\tan(b(x+c))+d\): period \(\pi/|b|\), vertical asymptotes where \(b(x+c)=\pi/2+k\pi\).

General solutions. \(\sin x=\sin\alpha\Rightarrow x=\alpha+2k\pi\) or \(x=\pi-\alpha+2k\pi\); \(\cos x=\cos\alpha\Rightarrow x=\pm\alpha+2k\pi\); \(\tan x=\tan\alpha\Rightarrow x=\alpha+k\pi\). Restrict to the requested interval.

Inverse trig. \(\arcsin:[-1,1]\to[-\pi/2,\pi/2]\); \(\arccos:[-1,1]\to[0,\pi]\); \(\arctan:\R\to(-\pi/2,\pi/2)\).