Solving trigonometric equations

Maths · Solutions of trigonometric equations. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A trigonometric equation contains a trigonometric function of the unknown angle. Because these functions are periodic, such an equation usually has many solutions. First find the solutions in one period, normally [0, 2π]. To do this, find the reference angle and then choose the quadrants where the function has the required sign.

The general solution adds every whole number of periods: 2nπ for sine and cosine, nπ for tangent, where n is any integer.

Harder equations are first reduced to simple ones. Bring everything to one side and factorise, or use an identity to get a quadratic in one function. Check any answers found after squaring.

Rules to remember

sin x = 0 gives x = nπ; cos x = 0 gives x = π/2 + nπ; tan x = 0 gives x = nπ (n ∈ Z)
sin x = k: if α is one solution, the other is π − α (add 2π if this is negative); general solution x = α + 2nπ or x = (π − α) + 2nπ
cos x = k: if α is one solution, the other in [0, 2π] is 2π − α; general solution x = α + 2nπ or x = (2π − α) + 2nπ
tan x = k: if α is one solution, the general solution is x = α + nπ
sin x and cos x lie between −1 and 1, so an equation such as sin x = 2 has no solution
A product equal to zero: put each factor equal to zero and solve each one
Quadratic form: use sin²x + cos²x = 1 to get one function only, then factorise

Common mistake

Dividing both sides by sin x or cos x throws away the solutions where that factor is zero. Bring all terms to one side and factorise instead.

Worked example

Solve 2 sin²x = 1 + cos x for x in [0, 2π].

Replace sin²x by 1 − cos²x: 2 − 2 cos²x = 1 + cos x, so 2 cos²x + cos x − 1 = 0.
Factorise: (2 cos x − 1)(cos x + 1) = 0, so cos x = 1/2 or cos x = −1.
cos x = 1/2 gives x = π/3 and 5π/3; cos x = −1 gives x = π.
Check x = π/3: 2 × (3/4) = 3/2 and 1 + 1/2 = 3/2.

Answer: x = π/3, π and 5π/3.

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