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AS Pure Mathematics · Edexcel 8MA0 · free A-level maths worksheet

Trigonometric identities and equations

sin squared plus cos squared equals 1, tan = sin/cos, solving in a given interval, equations in 2x or x + 30.

12 original questions, 51 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

Solve, for 0 ≤ x < 360°, the equation

sin x = 0.4

Give your answers to 1 decimal place.

2 marks
Show answer
x = 23.6°, 156.4°
2

Solve, for 0 ≤ x < 360°, the equation

tan x = −2

Give your answers to 1 decimal place.

3 marks
Show answer
x = 116.6°, 296.6°
3

Show that

1 − cos2 xcos x ≡ sin x tan x,   cos x ≠ 0

3 marks
Show answer
(1 - cos2 x)/cos x = sin2 x / cos x = sin x × (sin x / cos x) = sin x tan x.
4

Solve, for 0 ≤ x < 360°, the equation

2 cos(x + 30°) = 1

3 marks
Show answer
x = 30°, 270°
5

Solve, for 0 ≤ x < 360°, the equation

sin 2x = 0.6

Give your answers to 1 decimal place.

4 marks
Show answer
x = 18.4°, 71.6°, 198.4°, 251.6°
6

Solve, for 0 ≤ x < 360°, the equation

3 sin x = 2 cos x

Give your answers to 1 decimal place.

3 marks
Show answer
x = 33.7°, 213.7°
7

Given that sin θ = 25 and that θ is obtuse, find the exact value of

  1. cos θ
  2. tan θ, giving your answer with a rational denominator.
4 marks
Show answer
a) -√21/5 b) -2√21/21
8

Kofi is asked to solve, for 0 ≤ x < 360°, the equation

4 sin x cos x = 3 sin x

His working is

4 cos x = 3   so   cos x = 0.75   so   x = 41.4° or 318.6°

  1. Identify the error in Kofi's working.
  2. Find all the solutions of the equation in the interval, giving non-exact answers to 1 decimal place.
4 marks
Show answer
a) Dividing by sin x loses the solutions where sin x = 0 b) x = 0°, 41.4°, 180°, 318.6°
9
  1. Show that the equation 2 cos2 x + 3 sin x = 3 can be written as 2 sin2 x − 3 sin x + 1 = 0
  2. Hence solve, for 0 ≤ x < 360°, the equation 2 cos2 x + 3 sin x = 3
6 marks
Show answer
a) 2(1 - sin2 x) + 3 sin x = 3 rearranges to 2 sin2 x - 3 sin x + 1 = 0 b) x = 30°, 90°, 150°
10

Solve, for −180° ≤ x ≤ 180°, the equation

tan(2x − 15°) = √3

5 marks
Show answer
x = -142.5°, -52.5°, 37.5°, 127.5°
11
  1. Show that the equation 3 tan x sin x = 8 can be written as 3 cos2 x + 8 cos x − 3 = 0
  2. Hence solve, for 0 ≤ x < 360°, the equation 3 tan x sin x = 8, giving your answers to 1 decimal place.
  3. Hence find the solutions of 3 tan 2θ sin 2θ = 8 for 0 ≤ θ < 180°, giving your answers to 1 decimal place.
8 marks
Show answer
a) 3 sin2 x / cos x = 8 gives 3(1 - cos2 x) = 8 cos x, so 3 cos2 x + 8 cos x - 3 = 0 b) x = 70.5°, 289.5° c) θ = 35.3°, 144.7°
12

A passenger on a big wheel in Blackpool is at height h metres above the ground, t minutes after the ride starts, where

h = 20 − 18 cos(12t)°,   t ≥ 0

  1. Find the height of the passenger when the ride starts.
  2. Find the maximum height of the passenger and the time taken for one full turn of the wheel.
  3. Find the times during the first turn when the passenger is 29 m above the ground.
6 marks
Show answer
a) 2 m b) 38 m; 30 minutes c) 10 minutes and 20 minutes

Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.

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