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

Radians

Converting between radians and degrees, arc length, sector and segment area, small angle approximations, solving trig equations in radians.

12 original questions, 66 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
  1. Write 135° in radians, giving your answer as an exact multiple of π.
  2. Convert 2.4 radians to degrees, giving your answer to 1 decimal place.
  3. Find the exact value of sin 5π6
3 marks
Show answer
a) 3π4 b) 137.5° c) 12
2

A sector of a circle has radius 8 cm and angle 0.9 radians.

Find

  1. the length of the arc,
  2. the perimeter of the sector,
  3. the area of the sector.
4 marks
Show answer
a) 7.2 cm b) 23.2 cm c) 28.8 cm2
3

Given that θ is small and measured in radians, use the small angle approximations to show that

3 sin 2θ + cos 4θ ≈ a + bθ + cθ2

where a, b and c are integers to be found.

3 marks
Show answer
a = 1, b = 6, c = -8: 3 sin 2θ + cos 4θ ≈ 1 + 6θ - 8θ2
4

A sector of a circle has radius r cm and angle θ radians. The area of the sector is 20 cm2 and its perimeter is 18 cm.

Find the two possible pairs of values of r and θ.

5 marks
Show answer
a) r = 4, θ = 2.5 b) r = 5, θ = 1.6
5

The diagram shows a circle with centre O and radius 10 cm. The points A and B lie on the circle and angle AOB = 1.2 radians. The shaded region is the segment cut off by the chord AB.

  1. Find the area of the shaded segment.
  2. Find the perimeter of the shaded segment.

Give each answer to 3 significant figures.

O10 cm1.2 radABDiagram NOT accurately drawn
6 marks
Show answer
a) 13.4 cm2 b) 23.3 cm
6

Solve, for 0 ≤ x < 2π, the equation

2 cos2 x + 3 sin x = 3

giving your answers as exact multiples of π.

5 marks
Show answer
x = π/6, π/2, 5π/6
7

Solve, for 0 ≤ θ < π, the equation

3 cos(2θ + 0.5) = 2

giving your answers in radians to 3 significant figures.

5 marks
Show answer
θ = 0.171, 2.47
8

The equation sin θ + cos θ = 1.1 has a small positive root, θ = α, where θ is measured in radians.

  1. Use the small angle approximations to show that α is approximately a root of θ2 − 2θ + 0.2 = 0
  2. Hence find an approximate value for α, giving your answer to 4 decimal places and explaining your choice of root.
  3. Given that the true value of α is 0.10579 to 5 decimal places, find the percentage error in using 1 − √0.8 as an approximation for α.
6 marks
Show answer
a) shown b) 0.1056, taking 1 - √0.8 because the other root 1.894 is not small c) 0.20%
9

The diagram shows a sector OAB of a circle with centre O and radius 6 cm. Angle AOB = 0.8 radians. The tangent to the circle at A meets the line OB extended at the point C. The shaded region is bounded by AC, CB and the arc AB.

  1. Find the length of AC.
  2. Find the area of the shaded region.
  3. Find the perimeter of the shaded region.

Give each answer to 3 significant figures.

6 cm0.8 radOABCDiagram NOT accurately drawn
7 marks
Show answer
a) 6.18 cm b) 4.13 cm2 c) 13.6 cm
10

The points A and B lie on a circle with centre O and radius 5 cm, and angle AOB = θ radians, where 0 < θ < π. The chord AB divides the sector OAB into the triangle OAB and a segment. The area of the segment is equal to the area of the triangle OAB.

  1. Show that θ = 2 sin θ
  2. Show that this equation has a root between 1.8 and 2.0
  3. Given that the root is θ = 1.8955 to 4 decimal places, find the area of the segment, giving your answer to 3 significant figures.
7 marks
Show answer
a) shown b) f(θ) = θ - 2 sin θ: f(1.8) = -0.148 0, f is continuous, so there is a root c) 11.8 cm2
11

A flower bed is in the shape of the region between two arcs of circles with the same centre O, as shown in the diagram. The inner arc has radius r metres and the outer arc has radius 2r metres. Both arcs subtend an angle of θ radians at O, and θ > 1

The perimeter of the flower bed is 26 m and its area is 36 m2.

  1. Show that r2θ = 24
  2. Show that r2 − 13r + 36 = 0
  3. Find the value of r and the value of θ, explaining why you reject one solution.
rrθOrDiagram NOT accurately drawn
7 marks
Show answer
a) shown b) shown c) r = 4, θ = 1.5; r = 9 gives θ = 2481 ≈ 0.296 < 1, so it is rejected
12

A pendulum consists of a small bob on a light string of length L metres. When the string makes an angle θ radians with the vertical, the bob has moved a distance s metres along its circular path from the lowest point, and it is h metres above the lowest point, where

h = L(1 − cos θ)

  1. Write down s in terms of L and θ.
  2. Show that, when θ is small, h ≈ s22L

A pendulum with L = 2 is pulled back until the string makes an angle of 8° with the vertical.

  1. Use the approximation in part (b) to estimate h, and find the percentage error compared with the exact value of h. Give your answers to 2 significant figures.
  2. Explain why the approximation becomes less accurate as the angle increases.
8 marks
Show answer
a) s = Lθ b) shown c) h ≈ 0.019 m; percentage error 0.16% d) cos θ ≈ 1 - θ2/2 is only accurate when θ is small; the error grows quickly as θ increases

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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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