Trigonometric identities and equations

Maths: Analysis and Approaches · Geometry and trigonometry. 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

An identity is true for every angle. You use identities to change one ratio into another, to simplify expressions and to rewrite an equation so that it has only one ratio in it.

If you know one ratio and the quadrant of the angle, you can find the others. Use the Pythagorean identity for the size and the quadrant for the sign.

A trigonometric equation usually has more than one solution in the given interval. Find the first angle, then use the symmetry of the graph or the unit circle to find the rest. If the equation has sin²x or cos²x in it, use an identity to make a quadratic in one ratio, then factorise.

Rules to remember

sin²θ + cos²θ = 1 and tan θ = sin θ ÷ cos θ
sin 2θ = 2 sin θ cos θ
cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ
sin x = k: if one solution is α, another is π − α
cos x = k: if one solution is α, another is 2π − α
tan x = k: if one solution is α, another is α + π
For sin bx, cos bx or tan bx, find all values of bx in the interval multiplied by b, then divide each by b

Common mistake

Students divide both sides by sin x or cos x and lose solutions. Move everything to one side and factorise instead, so that sin x = 0 or cos x = 0 is kept as a possible solution.

Worked example

Solve sin 2x = sin x for 0 ≤ x ≤ π.

Use sin 2x = 2 sin x cos x: 2 sin x cos x − sin x = 0.
Factorise: sin x (2cos x − 1) = 0, so sin x = 0 or cos x = ½.
In the interval, sin x = 0 gives x = 0 and x = π; cos x = ½ gives x = π/3.

Answer: x = 0, π/3 and π.

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Topic 3: Geometry and Trigonometry

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