A LevelMathsAQAPureExam QuestionsTrigonometryRadian Measure

Radian Measure (AQA A Level Maths: Pure): Exam Questions

Exam code: 7357

2 hours30 questions
Easy
Medium
Hard
1
5 marks

Without using a calculator

(i) convert 60° into radians,

(ii) write down the exact numerical value of sin 45 degree,

(iii) write down the exact numerical value of tan pi ,

(iv) convert fraction numerator 5 pi over denominator 6 end fraction radians into degrees,

(v) write down the exact numerical value of cos 0.

How did you do?

2
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2 marks

A sector O P Q of a circle centre O has a radius of 4 cm and a central angle of pi over 4 radians.

(i) Find the length of the arc P Q.

(ii) Find the area of the sector O P Q.

How did you do?

3
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5 marks

Complete the table.

degrees

radians

sin

cos

tan

 

straight pi over 6

 

fraction numerator square root of 3 over denominator 2 end fraction

 

45 to the power of blank to the power of straight o end exponent

 

 

fraction numerator square root of 2 over denominator 2 end fraction

 

60 to the power of blank to the power of straight o end exponent

straight pi over 3

 

 

 

 

fraction numerator 2 straight pi over denominator 3 end fraction

fraction numerator square root of 3 over denominator 2 end fraction

 

 

270 to the power of blank to the power of straight o end exponent

 

 

 

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How did you do?

4
4 marks

Given that theta is small and measured in radians, use the small angle approximations to find an approximate value of

(i) sin theta plus cos theta

(ii) 2 tan squared theta minus cos theta

How did you do?

5
2 marks

The sector of a circle is shown below.

Sector OBC with radius 8 cm and angle π/3 radians at the centre O, labelled 'Not to scale'.

Find the perimeter of the sector.

Give your answer in the form p plus q pi cm where p and q are constants to be found.

How did you do?

6
2 marks

The arc length of a sector is 5 cm.

The radius is twice the arc length.

Find, in radians, the central angle of the sector.

How did you do?

7a
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2 marks

The diagram below shows the sector O P Q with radius 6 cm and central angle pi over 12 radians.

Diagram of a sector with centre O, radius 6 cm, angle π/12. Arc PQ forms a shaded segment on the sector's top edge.

Use the formula 1 half a b sin C to find the area of triangle O P Q, to 3 significant figures.

How did you do?

7b
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2 marks

Find the exact area of the sector O P Q.

How did you do?

7c
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1 mark

Find the area of the shaded segment, to 3 significant figures.

How did you do?

8
4 marks

The area of a sector is 8 pi cm2.

The arc length of the sector is 4 pi cm.

Find the radius and the central angle of the sector.

How did you do?

9
3 marks

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

sin 4 theta almost equal to 2 sin 2 theta cos 2 theta

How did you do?

10
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4 marks

Given that sin pi over 3 equals fraction numerator square root of 3 over denominator 2 end fraction, find all the solutions to the equation

sin theta equals fraction numerator square root of 3 over denominator 2 end fraction

where negative 2 pi less or equal than theta less or equal than 2 pi.

How did you do?

11
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4 marks

A sector O P Q of a circle centre O has a radius of 3.4 cm and a central angle of fraction numerator 3 pi over denominator 4 end fraction radians.

(i) Find the exact length of the arc P Q.

(ii) Find the exact area of the sector O P Q.

How did you do?

12
2 marks

Given that theta is small and measured in radians, use the small angle approximations to find an approximate value of

2 sin theta plus cos theta minus tan squared theta

How did you do?

13
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4 marks

The sector of a circle is shown below.

Diagram of sector OAB with radius OA of 9 cm, central angle π/5 radians, and arc AB. The sector is marked "Not to scale."

Find the exact area and the exact perimeter of the sector.

How did you do?

14
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3 marks

A plastic puzzle piece is in the form of a prism where

  • the uniform cross section of the prism is a sector

  • the radius of the sector is 8 cm

  • the central angle of the sector is 1.2 radians

  • the height of the puzzle piece is 2 cm

Diagram of a three-dimensional geometric shape with a sector of 1.2 radians atop a rectangle labelled 8 cm wide and 2 cm high. Not to scale.

Find the volume of the puzzle piece.

How did you do?

1
3 marks

Given that theta is small and is measured in radians, use the small angle approximations to find an approximate value of

fraction numerator 1 minus cos space 4 theta over denominator 2 theta sin space 3 theta end fraction

How did you do?

2
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3 marks

A parachutist, modelled by the point O below, is heading towards the ground using a parachute, modelled by the shaded segment.

The radii O A and O B are strings that connect the parachutist to the parachute.

Diagram of a sector with centre O, angle π/4 radians. The segment formed by the line AB and the curve AB is shaded. A note on the right states "Not to scale".

The area of sector O A B is  fraction numerator 81 pi over denominator 200 end fractionm2.

Find the length of one of the strings.

How did you do?

3a
3 marks

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

16 minus 7 sin x plus 2 tan 2 x minus 18 cos x almost equal to a x squared plus b x plus c

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

How did you do?

3b
2 marks

Hence find two approximate solutions to the equation

16 minus 7 sin x plus 2 tan 2 x minus 18 cos x equals 0

How did you do?

4a
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3 marks

Find all the solutions to the equation

cos theta equals 1 half

in the interval  negative 2 pi less or equal than theta less or equal than 2 pi, giving your answers in exact form.

How did you do?

4b
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5 marks

Find all the solutions to the equation

5 sin 3 x equals 1

in the interval  0 less or equal than x less or equal than pi, giving your answers to 3 significant figures.

How did you do?

5a
2 marks

The sector O A B is shown in the diagram below, where

  • the central angle of the sector is pi over 3 radians

  • the line segments O C and B C have lengths of 8 cm and pcm respectively

  • C D is parallel to A B

Triangle ABO with angle π/3 rad, side CO 8 cm, arc AB, CD parallel to line AB, shaded area inside trapezium ADCB, and the segment AB is shaded.

Find an expression for the exact area of sector O A B in terms of p.

How did you do?

5b
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2 marks

Find the exact area of triangle O C D.

How did you do?

5c
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3 marks

The shaded shape A B C D, where A B is the arc of the sector, has an area of

open parentheses fraction numerator 50 pi over denominator 3 end fraction minus 16 square root of 3 close parentheses cm2

Find the value of p.

How did you do?

6a
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3 marks

The diagram below shows the sector O A B.

Triangle OAB with sides 5.4 cm, angle O is 1.2 radians, shaded segment between arc AB and chord AB, diagram not to scale.

Find the area of the shaded segment, giving your answer to 3 significant figures.

How did you do?

6b
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2 marks

Find the perimeter of the sector O A B, to 2 decimal places.

How did you do?

7
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4 marks
q8a-5-4-radian-measure-a-level-only-edexcel-a-level-pure-maths-hard

The diagram shows a prism where

  • the uniform cross section of the prism is a sector

  • the radius of the sector is open parentheses 2 a minus 3 close parentheses cm

  • the central angle of the sector is 1.7 radians

  • the height of the prism is 4 cm

Given that the volume of the prism is 7.65 cm3, find the value of a.

How did you do?

8
3 marks

Given that theta is small and measured in radians, use the small angle approximations to find an approximate value of

fraction numerator sin squared space theta plus cos space theta over denominator tan space theta plus sin space theta end fraction

How did you do?

9
4 marks

A sector O P Q of a circle centre O has a radius of a cm and a central angle of a squared pi radians, where a is a non-zero constant.

The area of sector O P Q is three times the length of the arc P Q.

Find the value of a.

How did you do?

10
3 marks

Given that theta is small and measured in radians, use the small angle approximations to find and simplify an approximate value of

fraction numerator sin squared theta plus 2 cos squared open parentheses 2 theta close parentheses minus tan open parentheses 3 theta close parentheses over denominator sin open parentheses 4 theta close parentheses end fraction

How did you do?

1a
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3 marks

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

cos squared open parentheses 3 x close parentheses plus 1 fourth tan to the power of 4 open parentheses 3 x close parentheses

can be approximated by

1 plus a x squared plus b x to the power of 4

where a and b are constants to be found.

How did you do?

1b
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2 marks

Hence find three approximate solutions to the equation

cos squared open parentheses 3 x close parentheses plus 1 fourth tan to the power of 4 open parentheses 3 x close parentheses equals 1

How did you do?

1c
2 marks

Comment on the validity of your solutions in part (b).

How did you do?

2a
3 marks

The sector O A B is shown in the diagram below, where

  • the central angle of the sector is theta radians

  • the radii O A and O B are both equal to r

  • A D equals B C equals 1 cm

Diagram of a sector with centre O, radius r, angle θ, and shaded lens-shaped region ABCD. AB is an arc, and DC is a straight line which is a chord of the circle. OA=OB=r and OD =OC= r-1

Find an expression for the shaded area A B C D in terms of r and theta.

How did you do?

2b
3 marks

Show that, for values of theta near to zero, an approximation of the shaded area A B C D is given by the formula

1 half theta open parentheses a r plus b close parentheses

where a and b are constants to be found.

How did you do?

3a
3 marks

The diagram below shows the sector O A B.

Diagram of a sector with centre O, radius r cm, angle θ radians, and arc AB. The shaded area represents the segment.

Show that the area of the shaded segment is

1 half r squared open parentheses theta minus sin space theta close parentheses cm2

How did you do?

3b
2 marks

Find, in terms of theta, an expression for the percentage of the sector that the segment occupies.

How did you do?

4a
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5 marks

Given that theta is small and measured in radians, use the small angle approximations to find approximate solutions to the equation

fraction numerator 4 cos squared open parentheses 4 theta close parentheses minus sin squared open parentheses 2 theta close parentheses over denominator tan squared theta end fraction equals 12

How did you do?

4b
1 mark

Determine which out of the solutions in part (a) are more accurate, giving a reason for your answer.

How did you do?

5a
2 marks

A sector O S T of a circle centre O has a radius of r cm and a central angle of theta radians.

The chord S T has length a cm.

Show that 

a squared equals 2 r squared open parentheses 1 minus cos space theta close parentheses

How did you do?

5b
5 marks

Given that

  • r equals 4 theta

  • the area of the sector O S T is fraction numerator 8 pi cubed over denominator 27 end fraction cm2

find the exact value of a.

How did you do?

6
4 marks

The sector O A B is shown in the diagram below, where

  • the central angle of the sector is theta radians

  • the radii O A and O B are both equal to r

  • The line D C is perpendicular to the line O B

  • B C colon C O equals 2 colon 3

A geometric diagram showing a circular sector with center O, radius r cm, and angle θ radians. The arc extends from point B (bottom-left) to point A (top-left). A quadrilateral ABCD is inscribed within the sector, with B at the bottom-left on the arc, C on the base line between B and O, D directly above C on the line OA, and A at the top-left on the arc. The side BC lies along the base line, DC is vertical, and AB and AD follow the arc. The regions inside the sector but outside the quadrilateral are shaded gray. The base from C to O is labeled r cm, the line from D to O is labeled r cm, and the angle at O is labeled θ rad.

Show that the area of the shaded shape A B C D is

1 over 50 r squared open parentheses a straight theta plus b tan space theta close parentheses cm2 

where a and b are constants to be found.

How did you do?