Solving Trigonometric Equations

This lesson covers techniques for solving trigonometric equations at A-Level, including equations involving multiple angles, equations reducible to quadratics, equations requiring the use of identities, and finding general solutions. This is one of the most frequently examined topics in A-Level trigonometry.

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General Approach

1. Reduce to a single trigonometric function using identities if needed.
2. Find the principal value using inverse trig functions or exact values.
3. Find all solutions in the given range using the CAST diagram or symmetry.
4. Check your answers by substituting back.

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Basic Equations

Equations of the form sin x = k, cos x = k, tan x = k

Example 1: Solve sin x = 1/2 for 0° ≤ x ≤ 360°.

Principal value: x = arcsin(1/2) = 30°.

sin is positive in quadrants 1 and 2 (from CAST diagram):

x = 30° or x = 180° − 30° = 150°

Solutions: x = 30°, 150°

Example 2: Solve cos x = −√3/2 for 0 ≤ x ≤ 2π.