Trigonometry Reference
Trigonometric Functions and Identities
The six trigonometric functions describe ratios and unit-circle coordinates. Their domains and ranges tell you where each function is defined; identities then let you rewrite equivalent expressions.
Solved Examples
Jump directly to a worked example by identity or angle type.
Six-Function Domain and Range
| Function | Definition | Domain | Range |
|---|---|---|---|
| $\sin x$ | Unit-circle y-coordinate | $(-\infty,\infty)$ | $[-1,1]$ |
| $\cos x$ | Unit-circle x-coordinate | $(-\infty,\infty)$ | $[-1,1]$ |
| $\tan x=\frac{\sin x}{\cos x}$ | Undefined when $\cos x=0$ | $x\ne\frac\pi2+k\pi$ | $(-\infty,\infty)$ |
| $\cot x=\frac{\cos x}{\sin x}$ | Undefined when $\sin x=0$ | $x\ne k\pi$ | $(-\infty,\infty)$ |
| $\sec x=\frac1{\cos x}$ | Reciprocal of cosine | $x\ne\frac\pi2+k\pi$ | $(-\infty,-1]\cup[1,\infty)$ |
| $\csc x=\frac1{\sin x}$ | Reciprocal of sine | $x\ne k\pi$ | $(-\infty,-1]\cup[1,\infty)$ |
Here $k\in\mathbb Z$. Tangent and secant fail wherever cosine is zero; cotangent and cosecant fail wherever sine is zero.
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. |
Identity Tables
| Family | Identities |
|---|---|
| Reciprocal | $\sin x=\frac1{\csc x}$, $\cos x=\frac1{\sec x}$, $\tan x=\frac1{\cot x}$ |
| Quotient | $\tan x=\frac{\sin x}{\cos x}$, $\cot x=\frac{\cos x}{\sin x}$ |
| Pythagorean | $\sin^2x+\cos^2x=1$, $1+\tan^2x=\sec^2x$, $1+\cot^2x=\csc^2x$ |
| Even and odd | $\sin(-x)=-\sin x$, $\cos(-x)=\cos x$, $\tan(-x)=-\tan x$; reciprocals follow the same parity. |
| Cofunction | $\sin(\frac\pi2-x)=\cos x$, $\cos(\frac\pi2-x)=\sin x$, $\tan(\frac\pi2-x)=\cot x$ |
| 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 | $\sin2x=2\sin x\cos x$; $\cos2x=\cos^2x-\sin^2x=1-2\sin^2x=2\cos^2x-1$ |
| Half angle | $\sin^2\frac x2=\frac{1-\cos x}{2}$, $\cos^2\frac x2=\frac{1+\cos x}{2}$ |
The signs in half-angle formulas depend on the quadrant of $x/2$. For a proof, begin with the side that has more structure and preserve its domain restrictions.
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 Steps
- Rewrite secant as the reciprocal of cosine.
- Combine the terms over a common denominator.
- Replace $1-\cos^2x$ with $\sin^2x$.
- Rewrite one factor as tangent.
- State the verified identity.
$$\sec x-\cos x=\frac1{\cos x}-\cos x$$
$$\sec x-\cos x=\frac{1-\cos^2x}{\cos x}$$
$$\sec x-\cos x=\frac{\sin^2x}{\cos x}$$
$$\sec x-\cos x=\sin x\cdot\frac{\sin x}{\cos x}$$
$$\sec x-\cos x=\sin x\tan x$$
Exact Value
Find $\sin(75^\circ)$ exactly.
Solution Steps
- Rewrite $75^\circ$ as $45^\circ+30^\circ$.
- Apply the sine sum identity.
- Substitute the special-angle values.
- State the exact value: $\frac{\sqrt6+\sqrt2}{4}$.
$$\sin(75^\circ)=\sin(45^\circ+30^\circ)$$
$$\sin(75^\circ)=\sin45^\circ\cos30^\circ+\cos45^\circ\sin30^\circ$$
$$\sin(75^\circ)=\frac{\sqrt2}{2}\cdot\frac{\sqrt3}{2}+\frac{\sqrt2}{2}\cdot\frac12$$
$$\sin(75^\circ)=\frac{\sqrt6+\sqrt2}{4}$$
Double-Angle Rewrite
Rewrite $1-2\sin^2x$.
Solution Steps
- Choose the cosine double-angle form containing $\sin^2x$.
- State the equivalent expression: $\cos(2x)$.
$$\cos(2x)=1-2\sin^2x$$
$$1-2\sin^2x=\cos(2x)$$