Math · Precalculus · Concept

Trigonometric identities and equations

A trigonometric identity is an equation that holds at every angle where both sides are defined. The Pythagorean, sum and double-angle identities, such as cos 2x = 1 − 2sin²x, rewrite an expression into a more useful form: to simplify it, to prove another identity or to solve an equation. A trigonometric equation such as 2 sin x = 1, unlike an identity, holds only at particular angles, and usually at infinitely many of them.

Identity or equation?

sin²x + cos²x = 1 is true for every x: it is an identity. 2 sin x = 1 is true only at some angles: it is an equation to solve. One angle where a statement fails shows it is not an identity; showing that it is one takes an argument, because a table of values can only suggest it.

The Pythagorean identities

The point (cos x, sin x) lies on the unit circle, so cos²x + sin²x = 1. Dividing by cos²x or by sin²x gives two more forms, each valid where its denominator is not zero.

{sin}^{2} x + {cos}^{2} x = 1, 1 + {tan}^{2} x = {sec}^{2} x, 1 + {cot}^{2} x = {csc}^{2} x

\begin{array}{c} {sin}^{2} x + {cos}^{2} x = 1 \\ 1 + {tan}^{2} x = {sec}^{2} x \\ 1 + {cot}^{2} x = {csc}^{2} x \end{array}

Sum and difference identities

The sine of a sum is not the sum of the sines. The correct formulas mix both functions, and the signs in the cosine formula are the opposite of those inside the angle.

\begin{array}{l} 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 \end{array}

\begin{aligned} 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 \end{aligned}

Double-angle identities

Setting B = A in the sum identities gives the double-angle forms. Cosine has three versions: replace cos²x or sin²x with the Pythagorean identity to pick the one that suits the problem.

\begin{array}{l} sin 2 x = 2 sin x cos x \\ cos 2 x = {cos}^{2} x - {sin}^{2} x = 2 {cos}^{2} x - 1 = 1 - 2 {sin}^{2} x \end{array}

\begin{aligned} sin 2 x & = 2 sin x cos x \\ cos 2 x & = {cos}^{2} x - {sin}^{2} x \\ = 2 {cos}^{2} x - 1 \\ = 1 - 2 {sin}^{2} x \end{aligned}

Proving an identity

Work on one side only, usually the more complicated one, and turn it into the other side one step at a time. Moving terms across the equals sign assumes the identity you are trying to prove. Useful moves: rewrite everything in sines and cosines, factor, combine fractions and use a Pythagorean identity.

Solving a trigonometric equation

Isolate the trig function, find the solutions in one full turn, [0, 2π), with the unit circle, then add whole turns. Sine and cosine repeat every 2π, so each solution x₀ gives x₀ + 2πk for every integer k; tangent repeats every π.

sin x = \frac{1}{2} ⟹ x = \frac{π}{6} + 2 π k or \frac{5 π}{6} + 2 π k

\begin{array}{c} sin x = \frac{1}{2} \\ x = \frac{π}{6} + 2 π k or \frac{5 π}{6} + 2 π k \end{array}

Values that are not special

When the value is not on the special-angle table, inverse sine gives one solution and the unit circle gives the other. If sin x = 1/3, then x = arcsin(1/3) ≈ 0.3398, or x = π − 0.3398 ≈ 2.8018 in quadrant II.

x = arcsin \frac{1}{3} \approx 0.3398 or x \approx 2.8018

\begin{array}{c} x = arcsin \frac{1}{3} \approx 0.3398 \\ or x \approx 2.8018 \end{array}

Equations that factor

An equation that is quadratic in sin x or cos x factors like any quadratic. Set each factor equal to zero and solve each simpler equation. Dividing both sides by sin x instead would lose the solutions where sin x = 0.

Common mistakes

Writing sin(A + B) = sin A + sin B.
Dividing both sides by sin x or cos x, which loses the solutions where it is zero.
Stopping at the calculator’s one answer: sin x = 1/3 also has the solution π − arcsin(1/3) in [0, 2π).
Proving an identity by working on both sides at once, or by moving terms across the equals sign.

Key terms

Mathematical identity: An equation that is true for every allowed value of the variable, such as (x + 1)² = x² + 2x + 1. Testing a few numbers can disprove an identity but can’t prove one.
Pythagorean identity: The identity sin²θ + cos²θ = 1, true for every angle because (cos θ, sin θ) lies on the unit circle. Dividing it by cos²θ or by sin²θ gives 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ.
Double-angle formulas: Identities for trig functions of 2θ: sin 2θ = 2 sin θ cos θ and cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ. They are the sum formulas with both angles equal.
Unit circle: The circle of radius 1 centered at the origin. An angle θ in standard position meets it at the point (cos θ, sin θ), which defines sine and cosine for every real angle.
Inverse sine: arcsin x, or sin⁻¹ x: the angle between −π/2 and π/2 (−90° to 90°) whose sine is x, for −1 ≤ x ≤ 1. The equation sin θ = x has other solutions too; arcsin gives just this one.
Periodic function: A function that repeats after a fixed shift T: f(x + T) = f(x) for every x. The smallest such T is the period, such as 2π for sin x.

Work through an example

Prove that cos⁴x − sin⁴x = cos 2x.

Prove a trigonometric identity →

Solve a basic trigonometric equation →

Solve a trigonometric equation by factoring →

Use a double-angle identity →

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