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

Addition formulae, double angles and R sin(x + a)

Addition formulae, double angle formulae, expressing a cos x + b sin x as R sin(x + a), maxima and minima, equations and proofs, modelling.

12 original questions, 72 marks. Type your answers below for an instant mark and a score (algebra in any equivalent form, e.g. 2sqrt(3) or 2√3), or print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

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The questions

1

Without using a calculator, find the exact value of

  1. sin 7π12, giving your answer in the form √a + √b4
  2. tan π12, giving your answer in the form p + q√3 where p and q are integers.
4 marks
Show answer
a) √6 + √24 b) 2 - √3
2

Given that sin A = 45, where π2 < A < π, and cos B = 513, where 0 < B < π2, find the exact value of

  1. sin 2A
  2. cos(A − B)
5 marks
Show answer
a) -2425 b) 3365
3
  1. Express 5 sin x − 2 cos x in the form R sin(x − α), where R > 0 and 0 < α < π2
    Give the exact value of R and give the value of α in radians to 3 decimal places.
  2. Write down the maximum value of 5 sin x − 2 cos x
  3. Find the smallest positive value of x at which this maximum occurs, giving your answer to 3 significant figures.
5 marks
Show answer
a) √29 sin(x - 0.381) b) √29 c) x = 1.95
4

Prove that

cot θ − tan θ ≡ 2 cot 2θ,   θ ≠ nπ2, n ∈ ℤ

3 marks
Show answer
cot θ - tan θ = (cos²θ - sin²θ)/(sin θ cos θ) = cos 2θ/(½ sin 2θ) = 2 cot 2θ
5

In this question you must show all stages of your working.

Solve, for 0 ≤ x < 2π, the equation

cos 2x = 3 sin x − 1

Give your answers as exact multiples of π.

5 marks
Show answer
x = π/6, 5π/6
6

Solve, for 0 ≤ θ < 360°, the equation

5 sin 2θ = 3 cos θ

Give your answers to one decimal place where appropriate.

5 marks
Show answer
θ = 17.5°, 90°, 162.5°, 270°
7

Solve, for 0 ≤ θ < π, the equation

tan(θ + π4) = 6 tan θ

Give your answers in radians to 3 decimal places.

5 marks
Show answer
θ = 0.322, 0.464
8
  1. Show that sin 3x ≡ 3 sin x − 4 sin3 x
  2. Hence solve, for 0 < x < π, the equation
    sin 3x = 2 sin x
    giving your answers as exact multiples of π.
7 marks
Show answer
a) sin 3x ≡ 3 sin x - 4 sin3 x shown b) x = π6, 5π6
9
  1. Express 2 sin θ + 5 cos θ in the form R sin(θ + α), where R > 0 and 0 < α < 90°
    Give the exact value of R and give the value of α to 2 decimal places.
  2. Hence solve, for 0 ≤ θ < 360°, the equation
    2 sin θ + 5 cos θ = 3
    giving your answers to one decimal place.
6 marks
Show answer
a) √29 sin(θ + 68.20°) b) θ = 77.9°, 325.7°
10

In this question you must show all stages of your working.

  1. Express √3 sin x + cos x in the form R sin(x + α), where R > 0 and 0 < α < π2, giving the exact values of R and α.
  2. Hence solve, for 0 ≤ x < π, the equation
    √3 sin 2x + cos 2x = 1
  3. Find the maximum value and the minimum value of
    14 + √3 sin x + cos x
8 marks
Show answer
a) 2 sin(x + π6) b) x = 0, π3 c) maximum 12, minimum 16
11

The depth of water, D metres, at the entrance to a harbour is modelled by the equation

D = 5 + 1.2 sin(πt6) − 0.9 cos(πt6),   0 ≤ t < 12

where t is the time in hours after midnight and the angles are in radians.

  1. Express 1.2 sin x − 0.9 cos x in the form R sin(x − α), where R > 0 and 0 < α < π2
    Give the value of α to 4 decimal places.
  2. Using your answer to part (a), find the maximum depth of the water according to the model.
  3. Find the time when this maximum depth occurs, giving your answer to the nearest minute.
  4. A fishing boat can only pass through the entrance when the depth of water is at least 5.9 metres. Find, according to the model, the times between which the boat can pass through the entrance, giving your answers to the nearest minute.
9 marks
Show answer
a) 1.5 sin(x - 0.6435) b) 6.5 m c) 04:14 d) from 02:27 to 06:00
12

f(x) = 4 sin2 x + 6 sin x cos x,   0 ≤ x ≤ π

  1. Show that f(x) = 2 + 3 sin 2x − 2 cos 2x
  2. Express 3 sin 2x − 2 cos 2x in the form R sin(2x − α), where R > 0 and 0 < α < π2
    Give the exact value of R and give the value of α in radians to 4 decimal places.
  3. Hence state the maximum value of f(x) and find the value of x at which it occurs, giving x to 3 significant figures.
  4. Find the set of values of the constant k for which the equation f(x) = k has exactly two solutions in the interval 0 ≤ x ≤ π
10 marks
Show answer
a) shown b) √13 sin(2x - 0.5880) c) maximum 2 + √13 at x = 1.08 d) 2 - √13 < k < 2 + √13, k ≠ 0

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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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