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

Reciprocal trigonometric functions

sec, cosec and cot, their graphs, the identities 1 + tan² = sec² and 1 + cot² = cosec², inverse trig functions, proving identities and solving equations.

12 original questions, 58 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

Find the exact value of

  1. sec 3π4
  2. cosec π3
  3. cot 7π6
3 marks
Show answer
a) -√2 b) 2√33 c) √3
2

Solve, for 0 ≤ x < 2π, the equation

2 cosec x = 5

Give your answers to 3 significant figures.

3 marks
Show answer
x = 0.412, 2.73
3
  1. For y = arcsin x, state the largest possible domain and the corresponding range.
  2. Find the exact value of x for which arccos(2x − 1) = π3
4 marks
Show answer
a) domain -1 ≤ x ≤ 1; range -π2 ≤ y ≤ π2 b) x = 34
4

Given that tan θ = −2, where π2 < θ < π, find the exact value of

  1. sec θ
  2. cosec θ
4 marks
Show answer
a) sec θ = -√5 b) cosec θ = √52
5

Prove that

sec2 θ + cosec2 θ ≡ sec2 θ cosec2 θ

3 marks
Show answer
sec²θ + cosec²θ = (sin²θ + cos²θ)/(sin²θ cos²θ) = 1/(sin²θ cos²θ) = sec²θ cosec²θ.
6

Solve, for 0 ≤ x < 360°, the equation

tan2 x = 2 sec x + 2

Give your answers to 1 decimal place where appropriate.

(Solutions based entirely on graphical or numerical methods are not acceptable.)

5 marks
Show answer
x = 70.5°, 180°, 289.5°
7

Solve, for 0 ≤ θ < 2π, the equation

3 cot2 θ − 5 cosec θ + 1 = 0

Give your answers as exact multiples of π.

5 marks
Show answer
θ = π/6, 5π/6
8

Solve, for 0 ≤ x < 2π, the equation

cosec x = 4 cos x

Give your answers as exact multiples of π.

4 marks
Show answer
x = π/12, 5π/12, 13π/12, 17π/12
9
  1. Sketch the graph of y = 3 − 2 sec x for 0 ≤ x ≤ 2π. Show the coordinates of the end points and of any turning points, and the equations of any asymptotes.
  2. Hence state the set of values of k for which the equation 3 − 2 sec x = k has no solutions.
0123456−6−4−2246810xy
6 marks
Show answer
a) asymptotes x = π2 and x = 3π2; end points (0, 1) and (2π, 1); minimum point (π, 5) b) 1 < k < 5
10

The function f is defined by

f(x) = π − 2 arccos(x3)

  1. State the largest possible domain of f.
  2. Find the range of f.
  3. Solve the equation f(x) = π3
5 marks
Show answer
a) -3 ≤ x ≤ 3 b) -π ≤ f(x) ≤ π c) x = 32
11
  1. Prove that
    1 + sec θtan θ + sin θ ≡ cosec θ,   θ ≠ nπ2, n ∈ ℤ
  2. Hence solve, for 0 < x < 2π, the equation
    1 + sec xtan x + sin x + cot2 x = 5
    Give exact answers where possible and other answers to 3 significant figures.
8 marks
Show answer
a) (1 + sec θ)/(tan θ + sin θ) = ((1 + cos θ)/cos θ) ÷ (sin θ(1 + cos θ)/cos θ) = 1/sin θ = cosec θ b) x = π6, 5π6, 3.48, 5.94
12

A ladder rests on horizontal ground at A and leans against a vertical wall at B. On its way it rests on the top edge F of a fence. The fence is 2 m tall, parallel to the wall and 1 m in front of it. The ladder makes an angle θ radians with the ground, where 0 < θ < π2. The ladder is modelled as a straight rod AB of length L metres that just touches F.

  1. Show that L = 2 cosec θ + sec θ
  2. Show that dLdθ = 0 when tan3 θ = 2
  3. Given that this gives the shortest ladder that can reach the wall over the fence, find this shortest length. Give your answer to 3 significant figures.
8 marks
Show answer
a) AF = 2 cosec θ and FB = sec θ, so L = 2 cosec θ + sec θ b) dL/dθ = -2 cosec θ cot θ + sec θ tan θ = 0 gives sin3 θ = 2 cos3 θ, so tan3 θ = 2 c) L = 4.16 m

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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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