Trigonometry visual reference

Trigonometric Identities Cheat Sheet

Trig identities are easier to use when they are grouped by purpose. Start with reciprocal, quotient, and Pythagorean identities; use the other groups to rewrite angles or change the form of an expression.

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Trigonometric identities cheat sheet organized into reciprocal quotient Pythagorean even odd cofunction and double angle formula groups
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How to read it

Use the chart as a decision tool.

A printable trig identities cheat sheet organized by reciprocal, quotient, Pythagorean, even-odd, cofunction, and double-angle identities. Read the label first, identify the matching structure in your problem, then confirm that any stated conditions apply before substituting values.

01

Reciprocal and quotient

Rewrite everything in sine and cosine when you are stuck.

Reciprocal

csc⁡θ=1sin⁡θ,sec⁡θ=1cos⁡θ\displaystyle \csc\theta=\frac1{\sin\theta},\quad\sec\theta=\frac1{\cos\theta}

Quotient

tan⁡θ=sin⁡θcos⁡θ,cot⁡θ=cos⁡θsin⁡θ\displaystyle \tan\theta=\frac{\sin\theta}{\cos\theta},\quad\cot\theta=\frac{\cos\theta}{\sin\theta}
02

Pythagorean and symmetry

These connect squares and negative angles.

Pythagorean

sin⁡2θ+cos⁡2θ=1\displaystyle \sin^2\theta+\cos^2\theta=1

Tangent form

1+tan⁡2θ=sec⁡2θ\displaystyle 1+\tan^2\theta=\sec^2\theta

Even / odd

cos⁡(−θ)=cos⁡θ,sin⁡(−θ)=−sin⁡θ\displaystyle \cos(-\theta)=\cos\theta,\quad\sin(-\theta)=-\sin\theta
03

Angle changes

Use these when the angle itself must be rewritten.

Cofunction

sin⁡(π2−θ)=cos⁡θ\displaystyle \sin\left(\frac\pi2-\theta\right)=\cos\theta

Double-angle sine

sin⁡(2θ)=2sin⁡θcos⁡θ\displaystyle \sin(2\theta)=2\sin\theta\cos\theta

Double-angle cosine

cos⁡(2θ)=cos⁡2θ−sin⁡2θ\displaystyle \cos(2\theta)=\cos^2\theta-\sin^2\theta
How to remember it

Build from the three foundational families

  1. 1Reciprocal identities come from flipping a fraction.
  2. 2Divide sin²θ + cos²θ = 1 by cos²θ or sin²θ to produce the other Pythagorean identities.
  3. 3For cofunctions, complementary angles swap partners.
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