Trigonometry Reference
Trigonometric Identities
A trigonometric identity is an equation that is true for every allowed angle. The usual move is to rewrite the more complicated side until both sides match.
Fact Table
| Identity Family | Best Use | Common Move |
|---|---|---|
| Reciprocal identities | Convert secant, cosecant, and cotangent. | Rewrite in sine and cosine. |
| Quotient identities | Handle tangent and cotangent. | Use $\tan x=\frac{\sin x}{\cos x}$. |
| Pythagorean identities | Replace squares. | Use $\sin^2x+\cos^2x=1$ and its divided forms. |
| Even-odd identities | Simplify negative angles. | Cosine and secant are even; sine, tangent, and cotangent are odd. |
| Sum and difference identities | Exact values and angle expansion. | Split angles into familiar special angles. |
| Double-angle identities | Expressions with $2x$. | Choose the form that matches the problem. |
Content Formulas
Reciprocal and Quotient
$$\csc x=\frac1{\sin x}\quad \sec x=\frac1{\cos x}\quad \cot x=\frac1{\tan x}$$ $$\tan x=\frac{\sin x}{\cos x}\quad \cot x=\frac{\cos x}{\sin x}$$
Pythagorean
$$\sin^2x+\cos^2x=1$$ $$1+\tan^2x=\sec^2x$$ $$1+\cot^2x=\csc^2x$$
Sum and Difference
$$\sin(a\pm b)=\sin a\cos b\pm\cos a\sin b$$ $$\cos(a\pm b)=\cos a\cos b\mp\sin a\sin b$$
Double Angle
$$\sin(2x)=2\sin x\cos x$$ $$\cos(2x)=\cos^2x-\sin^2x$$ $$\tan(2x)=\frac{2\tan x}{1-\tan^2x}$$
When proving an identity, work on one side at a time. Avoid moving terms across the equals sign unless you are solving an equation.
Classic Examples
Verify a Basic Identity
Verify $\sec x-\cos x=\sin x\tan x$.
Solution StrategyRewrite the left side using sine and cosine, then combine fractions.
$$\sec x-\cos x=\frac1{\cos x}-\cos x$$
$$\sec x-\cos x=\frac{1-\cos^2x}{\cos x}$$
$$\sec x-\cos x=\frac{\sin^2x}{\cos x}$$
$$\sec x-\cos x=\sin x\cdot\frac{\sin x}{\cos x}$$
$$\sec x-\cos x=\sin x\tan x$$
Exact Value
Find $\sin(75^\circ)$ exactly.
Solution StrategyWrite $75^\circ$ as $45^\circ+30^\circ$.
$$\sin(75^\circ)=\sin(45^\circ+30^\circ)$$
$$\sin(75^\circ)=\sin45^\circ\cos30^\circ+\cos45^\circ\sin30^\circ$$
$$\sin(75^\circ)=\frac{\sqrt2}{2}\cdot\frac{\sqrt3}{2}+\frac{\sqrt2}{2}\cdot\frac12$$
$$\sin(75^\circ)=\frac{\sqrt6+\sqrt2}{4}$$
Double-Angle Rewrite
Rewrite $1-2\sin^2x$.
Solution StrategyChoose the double-angle cosine form that contains $\sin^2x$.
$$\cos(2x)=1-2\sin^2x$$
$$1-2\sin^2x=\cos(2x)$$