Trigonometric Identities Quick Reference Cheat Sheet

Key Facts

• On the unit circle, a point at angle $\theta$ has coordinates $(\cos \theta, \sin \theta)$.
• The tangent ratio is $\tan \theta = \frac{\sin \theta}{\cos \theta}$ when $\cos \theta \ne 0$.
• The main Pythagorean identity is $\sin^2 \theta + \cos^2 \theta = 1$.
• Dividing $\sin^2 \theta + \cos^2 \theta = 1$ by $\cos^2 \theta$ gives $1 + \tan^2 \theta = \sec^2 \theta$.
• Dividing $\sin^2 \theta + \cos^2 \theta = 1$ by $\sin^2 \theta$ gives $\cot^2 \theta + 1 = \csc^2 \theta$.
• The sine sum and difference formulas are $\sin(a + b) = \sin a \cos b + \cos a \sin b$ and $\sin(a - b) = \sin a \cos b - \cos a \sin b$.
• The cosine sum and difference formulas are $\cos(a + b) = \cos a \cos b - \sin a \sin b$ and $\cos(a - b) = \cos a \cos b + \sin a \sin b$.
• The double-angle formulas include $\sin(2\theta) = 2\sin \theta \cos \theta$, $\cos(2\theta) = \cos^2 \theta - \sin^2 \theta$, and $\tan(2\theta) = \frac{2\tan \theta}{1 - \tan^2 \theta}$.

Vocabulary

Unit circle

The unit circle is the circle with radius $1$ centered at the origin, used to define trigonometric values for all angles.

Trigonometric identity

A trigonometric identity is an equation involving trig functions that is true for every angle where both sides are defined.

Pythagorean identity

A Pythagorean identity is a trig equation derived from $\sin^2 \theta + \cos^2 \theta = 1$.

Reference angle

A reference angle is the acute angle formed between the terminal side of an angle and the $x$-axis.

Double-angle formula

A double-angle formula rewrites a trig function of $2\theta$ using trig functions of $\theta$.

Quadrant sign

A quadrant sign tells whether $\sin \theta$, $\cos \theta$, or $\tan \theta$ is positive or negative based on the quadrant of $\theta$.