Solving trigonometric equations

Step by step

1. Find the principal value with the inverse function on your calculator.

2. Use symmetry to find the second value:

• sin x = k: x and 180° - x
• cos x = k: x and 360° - x
• tan x = k: x and x + 180°

3. Add or subtract 360° (or 180° for tan) to find all solutions in the interval.

Multiples and shifts

For sin 2x = k with 0 ≤ x < 360°, solve for 2x in 0 ≤ 2x < 720°, then halve. For cos(x + 30°) = k with 0 ≤ x < 360°, solve for x + 30° in 30° ≤ x + 30° < 390°, then subtract 30°. Change the interval first.

Radians and negative intervals

If the interval is in radians, give answers in radians: the symmetries become π - x, 2π - x and x + π. For intervals such as -180° ≤ x ≤ 180°, subtract 360° from solutions that are too large.

Worked example

Solve cos 2x = 0.5 for 0 ≤ x < 360°.

1. Let 2x run over 0 ≤ 2x < 720°.
2. cos^-1(0.5) = 60°, and 360° - 60° = 300°
3. Add 360°: 420° and 660°
4. Halve: x = 30°, 150°, 210°, 330°

Answer: x = 30°, 150°, 210°, 330°

Check yourself

1. Solve cos x = -0.5 for 0 ≤ x < 360°.
   Show answer

   x = 120° or x = 240°. cos^-1(-0.5) = 120°. The second solution is 360° - 120° = 240°.

2. Solve tan x = 1 for 0 ≤ x < 360°.
   Show answer

   x = 45° or x = 225°. tan^-1(1) = 45°. tan repeats every 180°, so 225° too.

3. sin x = 0.3 for 0 ≤ x < 360°. One solution is 17.5° (1 d.p.). Find the other to 1 decimal place.
   Show answer

   162.5 °. 180° - 17.46° = 162.5°.

4. Solve sin 2x = sqrt(3)/2 for 0 ≤ x < 360°.
   Show answer

   x = 30°, 60°, 210° or 240°. For 0 ≤ 2x < 720°: 2x = 60°, 120°, 420°, 480°. Halve each.

5. Solve cos(x + 30°) = 0.5 for 0 ≤ x < 360°.
   Show answer

   x = 30° or x = 270°. For 30° ≤ x + 30° < 390°: x + 30° = 60° or 300° (420° is too large). Subtract 30°.

6. Solve sin x = 1/2 for 0 ≤ x < 2π.
   • x = π/6 or x = 7π/6
   • x = π/6 or x = 5π/6
   • x = π/3 or x = 2π/3
   • x = π/6 or x = 11π/6
   Show answer

   x = π/6 or x = 5π/6. sin^-1(1/2) = π/6, and π - π/6 = 5π/6.

7. Solve tan x = -1 for -180° ≤ x ≤ 180°.
   Show answer

   x = -45° or x = 135°. tan^-1(-1) = -45°. Adding 180° gives 135°.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 2 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.