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Higher Mathematics

Solving trigonometric equations

Solve equations over a given interval using symmetry and graph knowledge.

Topic practiceFormula supportMixed revision

Before you start

  • Know exact values for 0°, 30°, 45°, 60° and 90°.
  • Understand CAST or graph symmetry.
  • Check the interval stated in the question.
Higher Mathematics lesson

Explanation

Trig equations usually have more than one solution in a given interval. Find the related acute angle, then use symmetry or graphs to list all valid solutions.

Always finish by checking the interval and the original equation.

Visual support

periodamplitudeθ

Method and rules

  • sin x positive in quadrants I and II.
  • cos x positive in quadrants I and IV.
  • tan x positive in quadrants I and III.
  • sin² x + cos² x = 1

Worked examples

Worked example 1

Solve a sine equation

Solve 2sin x = 1 for 0° ≤ x < 360°.

  1. sin x = 12
  2. Reference angle is 30°.
  3. Sine is positive in quadrants I and II.

Answer: x = 30°, 150°

Worked example 2

Solve a cosine equation

Solve cos x = −12 for 0° ≤ x < 360°.

  1. Reference angle is 60°.
  2. Cosine is negative in quadrants II and III.

So: x = 120°, 240°

Worked example 3

Solve a tan equation

Solve tan x = 1 for 0° ≤ x < 360°.

  1. Reference angle is 45°.
  2. Tangent is positive in quadrants I and III.

Answer: x = 45°, 225°

Watch out

  • Giving only the calculator's first answer.
  • Including endpoints that are not in the interval.
  • Losing solutions after squaring or rearranging.

Exam reminder

List every solution in the stated interval and check whether endpoints are included. Keep exact values when exact angles are expected.