Trig identities and equations

Trig identities are equations true for every angle. The two most useful are tan θ = sin θ / cos θ and sin²θ + cos²θ = 1. They simplify expressions and help solve trig equations.

Key facts

  • tan θ = sin θ / cos θ
  • sin²θ + cos²θ = 1
  • CAST diagram: which ratios are positive in each quadrant.
  • Solutions repeat: sin and cos every 360°, tan every 180°.
  • Reduction formulas (CAPS): sin(180° − θ) = sin θ, cos(180° − θ) = −cos θ.

Worked examples

Example 1

Simplify (1 − cos²θ)/sin θ.

  1. 1 − cos²θ = sin²θ
  2. sin²θ / sin θ = sin θ

Answer: sin θ

Example 2

Solve 2 sin²θ = sin θ for 0° ≤ θ ≤ 360°.

  1. 2 sin²θ − sin θ = 0 sin θ(2 sin θ − 1) = 0
  2. sin θ = 0: θ = 0°, 180°, 360°.
  3. sin θ = ½: θ = 30°, 150°.

Answer: 0°, 30°, 150°, 180°, 360°