Trigonometric identities & equations

Overview

Identities let you simplify trig expressions; then you solve equations within a given range.

Key identities

• sin²θ + cos²θ = 1
• tan θ = sin θ ÷ cos θ
• 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ

Solving trig equations

1. Rearrange to the form sin θ = k (or cos/tan).
2. Find the first solution (the principal value).
3. Use the graph or symmetry to find all solutions in the stated range.

Worked example

Solve 2 sin θ = 1 for 0° ≤ θ ≤ 360°: sin θ = 0.5 → θ = 30°, and by symmetry θ = 180° − 30° = 150°. So θ = 30° and 150°.

Common mistakes

• Giving only one answer — sine and cosine usually have two solutions per cycle.

Exam tips

• Sketch the graph and draw the line y = k to count how many solutions to expect.