Trigonometry

AS Level Mathematics · Pure Mathematics 1. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.

Test this topic freeAll AS Level Mathematics topics

The idea

Sine, cosine and tangent are defined for angles of any size, and their graphs repeat. Sine and cosine repeat every 360° (2π radians) and stay between −1 and 1. Tangent repeats every 180° (π radians).

Because the graphs repeat, a trigonometric equation usually has more than one solution in the given interval. Your calculator gives only the principal value. You find the others from the symmetry of the graph or from the quadrant rule.

You must know the exact values for 30°, 45° and 60°, and use them for related angles such as 150° or 240°. Two identities let you simplify expressions and turn an equation into one that has a single trigonometric function.

Rules to remember

Common mistake

For an equation in 2x, students find 2x only between 0° and 360° and lose half the solutions. Double the interval first (0° to 720° for 2x), list every value of 2x, then halve them.

Worked example

Solve 2 cos2x + 3 sin x = 3 for 0° ≤ x ≤ 360°.

  1. Replace cos2x with 1 − sin2x: 2 − 2 sin2x + 3 sin x = 3.
  2. Rearrange: 2 sin2x − 3 sin x + 1 = 0, so (2 sin x − 1)(sin x − 1) = 0.
  3. sin x = 1/2 gives x = 30° and 180° − 30° = 150°. sin x = 1 gives x = 90°.

Answer: x = 30°, 90°, 150°

Practice questions

Try each one, then open the answer.

1. Solve 2 cos x + 1 = 0 for 0° ≤ x ≤ 360°.

  1. A
    120°, 240°
  2. B
    60°, 300°
  3. C
    60°, 120°
  4. D
    120° only
Show answer

Answer: A. cos x = −1/2. The related acute angle is 60°, and cosine is negative in the second and third quadrants: x = 180° − 60° = 120° and x = 180° + 60° = 240°.

2. Solve 3 sin x = 2 cos x for 0° ≤ x ≤ 360°, giving answers to 1 decimal place.

  1. A
    33.7°, 146.3°
  2. B
    56.3°, 236.3°
  3. C
    33.7° only
  4. D
    33.7°, 213.7°
Show answer

Answer: D. Divide by cos x: tan x = 2/3, so x = tan−1(2/3) = 33.7°. Tangent has period 180°, so the second solution is 33.7° + 180° = 213.7°.

3. Simplify (1 − cos2θ)/(sin θ cos θ).

  1. A
    sin θ
  2. B
    sin θ cos θ
  3. C
    cos θ/sin θ
  4. D
    tan θ
Show answer

Answer: D. Use sin2θ + cos2θ = 1: the numerator is sin2θ. Then sin2θ/(sin θ cos θ) = sin θ/cos θ = tan θ.

More questions on this topic

Read the full notes

Keep going

← Circular measureSeries →All rules on one pageStuck? Ask a question