Trig Identities and Equations: The Six That Solve Everything

The three Pythagorean identities

• sin²θ + cos²θ = 1 (the foundation)
• 1 + tan²θ = sec²θ (divide by cos²)
• 1 + cot²θ = csc²θ (divide by sin²)

Double-angle formulas

• sin(2θ) = 2 sin θ cos θ
• cos(2θ) = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ (three equivalent forms)
• tan(2θ) = 2 tan θ / (1 − tan²θ)

Solving trig equations: the multi-solution trap

When solving sin θ = c on [0, 2π], there are usually TWO solutions. Use the reference angle:

• sin positive → Q1 and Q2: θ = arcsin(c) and π − arcsin(c)
• sin negative → Q3 and Q4: θ = π + ref and 2π − ref

For sin(nθ) = c, solve for nθ first, then divide — and remember nθ spans [0, 2πn], so there are 2n solutions.

Quadratic in sin/cos

Equations like 2sin²x − sin x − 1 = 0 are quadratics in u = sin x. Factor: (2u + 1)(u − 1) = 0 → sin x = 1 or sin x = −1/2. Then solve each in the standard way.

Check yourself

Practice with real CBE questions

Pre-Calc Sem B practice for trig equations.