Revision NotesExam QuestionsPast PapersMock Exams
A LevelMathsCambridge (CIE)Pure 1Exam QuestionsTrigonometryTrigonometric Equations

Trigonometric Equations (Cambridge (CIE) A Level Maths: Pure 1): Exam Questions

Exam code: 9709

3 hours36 questions
Easy
Medium
Hard
Very Hard
1a
Sme Calculator2 marks

Work out the length of the missing side in the following right-angled triangle.

Right-angled triangle, not to scale, with sides 12 cm and 13 cm. Angle θ marked between the 12 cm side and hypotenuse.

How did you do?

1b
Sme Calculator3 marks

Using your answer from part (a) to help, write down the values of the following:

left parenthesis straight i right parenthesis space space sin space theta space space left parenthesis ii right parenthesis space space cos space theta space space left parenthesis iii right parenthesis space tan space theta

How did you do?

Did this page help you?

2
Sme Calculator2 marks

Show that

fraction numerator 1 minus cos squared x over denominator tan squared x end fraction identical to cos squared x

How did you do?

Did this page help you?

3
Sme Calculator3 marks

Solve the equation

sin x equals 1 half comma    0 degree less or equal than x less or equal than 360 degree

How did you do?

Did this page help you?

4a
Sme Calculator2 marks

Solve the equation x squared plus x minus 2 equals 0

How did you do?

4b
Sme Calculator2 marks

Hence, or otherwise, solve the equation cos squared x plus cos x minus 2 equals 0    text for  end text 0 degree less or equal than x less or equal than 720 degree

How did you do?

Did this page help you?

5
Sme Calculator3 marks

Solve the equation tan 2 theta equals 0.3    text for  end text minus 180 degree less or equal than theta less or equal than 180 degree comma giving your answers to one decimal place.

How did you do?

Did this page help you?

6a
Sme Calculator2 marks

Sketch the graph of y equals cos 2 x    text for  end text 0 less or equal than x less or equal than 2 pi.

How did you do?

6b
Sme Calculator2 marks

Solve the equation cos 2 x equals 0.5    text for  end text 0 less or equal than x less or equal than 2 pi.

How did you do?

Did this page help you?

7
Sme Calculator4 marks

Solve the equation 2 left parenthesis 1 minus cos squared theta right parenthesis equals 1    text for  end text minus pi less or equal than theta less or equal than pi.

How did you do?

Did this page help you?

8
Sme Calculator4 marks

Solve the equation 4 minus 4 sin squared theta equals 3    text for  end text 0 degree less or equal than theta less or equal than 180 degree.

How did you do?

Did this page help you?

9
Sme Calculator4 marks

Find all the solutions to the equation 2 sin theta equals square root of 3    text for  end text minus 2 pi less or equal than theta less or equal than 2 pi.

How did you do?

Did this page help you?

1a
Sme Calculator4 marks

Find all solutions to the equation cos space theta equals 1 half in the interval negative 2 pi less or equal than theta less or equal than 2 pi comma giving your answers in radians as multiples of pi.

How did you do?

1b
Sme Calculator6 marks

Find all 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 answer in radians to three significant figures.

How did you do?

Did this page help you?

2a
Sme Calculator2 marks

Show that the equation 2 sin squared x plus 3 cos x equals 0 can be written in the form   a cos squared x plus b cos x plus c equals 0 comma where a comma b text  and  end text c are integers to be found.

How did you do?

2b
Sme Calculator3 marks

Hence, or otherwise, solve the equation 2 sin squared x plus 3 cos x equals 0 for negative 180 degree less or equal than x less or equal than 180 degree

How did you do?

Did this page help you?

3
Sme Calculator3 marks

Given that sin space theta equals 3 over 5 find the possible values of cos space theta text  and  end text tan text end text theta.

How did you do?

Did this page help you?

4
Sme Calculator3 marks

Solve the equation 2 space sin 2 theta equals 1 for 0 less or equal than theta less or equal than 2 pi

How did you do?

Did this page help you?

5
Sme Calculator5 marks

Solve the equation 2 sin x equals fraction numerator 1 over denominator sin x end fractionfor 0 degree less or equal than x less or equal than 360 degree.

How did you do?

Did this page help you?

6
Sme Calculator6 marks

A right-angled triangle has hypotenuse 8 cm. One of its other sides is 5 cm.
Find exact values for sin space theta comma cos space theta text  and  end text tan text end text theta comma where theta is the smallest angle in the triangle.

How did you do?

Did this page help you?

7
Sme Calculator5 marks

Solve the equation 2 space sin space x space cos space x equals cos space x for negative pi less or equal than x less or equal than pi

How did you do?

Did this page help you?

8a
Sme Calculator2 marks

Show that left parenthesis x plus 1 right parenthesis left parenthesis x minus 2 right parenthesis left parenthesis x minus 3 right parenthesis identical to x cubed minus 4 x squared plus x plus 6.

How did you do?

8b
Sme Calculator5 marks

Hence, or otherwise, solve the equation tan cubed x minus 4 tan squared x plus tan x plus 6 equals 0 for 0 degree less or equal than x less or equal than 360 degree comma giving your answers to 1 decimal place where appropriate.

How did you do?

Did this page help you?

9a
Sme Calculator2 marks

A seagull sits on the surface of the sea and moves up and down as waves pass.

Its height, h metres, above its position in calm water is modelled by the function h equals 1 half sin left parenthesis 180 t right parenthesis where t is the time in seconds after timing commences.

Sketch a graph of h against t for 0 less or equal than t less or equal than 10 showing the coordinates of the points of intersection with the t axis.

How did you do?

9b
Sme Calculator1 mark

How many times in the first minute after timing commences is the seagull 0.25 metres above its calm water position?

How did you do?

9c
Sme Calculator3 marks

Find the time at which the seagull is first 0.25 m above its calm water position and moving downwards. Give your answer to 3 significant figures.

How did you do?

Did this page help you?

1
Sme Calculator3 marks

Solve the equation 2 space sin theta equals 3 space cos theta for 0 less or equal than theta less or equal than 2 pi comma giving your answers to 3 significant figures.

How did you do?

Did this page help you?

2
Sme Calculator5 marks

Solve the equation 2 sin squared theta equals cos theta plus 1for negative 180 degree less or equal than theta less or equal than 180 degree.

How did you do?

Did this page help you?

3
Sme Calculator3 marks

Given that the angle θ is obtuse and that sin space theta equals 3 over 4 comma find the exact value of cos theta.

How did you do?

Did this page help you?

4
Sme Calculator5 marks

Solve the equation tan 2 x equals fraction numerator 3 over denominator tan 2 x end fraction for negative 180 degree less or equal than x less or equal than 180 degree

How did you do?

Did this page help you?

5
Sme Calculator5 marks

Solve the equation 2 tan x minus sin x equals 0 for negative pi less or equal than x less or equal than pi

How did you do?

Did this page help you?

6
Sme Calculator6 marks

An isosceles triangle has sides 8 cm, 8 cm and 4 cm and equal base angles θ.

Find exact values for sin space theta comma space cos space theta space and space tan space theta.

How did you do?

Did this page help you?

7a
Sme Calculator4 marks

Find all the solutions to the equation square root of 3 tan 2 theta equals negative 1 in the interval negative pi less or equal than theta less or equal than pi , giving your answers in radians as multiples ofpi.

How did you do?

7b
Sme Calculator5 marks

Find all the solutions to the equation 6 sin squared x plus 7 sin x minus 3 equals 0 in the interval 0 less or equal than x less or equal than 2 pi , giving your answers in radians to three significant figures.

How did you do?

Did this page help you?

8a
Sme Calculator1 mark

Show that x equals 1 half satisfies the equation 8 x cubed minus 4 x squared minus 6 x plus 3 equals 0.

How did you do?

8b
Sme Calculator6 marks

Hence solve the equation 8 cos cubed x minus 4 cos squared x minus 6 cos x plus 3 equals 0 for 0 degree less or equal than x less or equal than 360 degree.

How did you do?

Did this page help you?

9a
Sme Calculator4 marks

A seagull sits on the surface of the sea and moves up and down as waves pass.

Its height, h metres, above its position in calm water is modelled by the function h equals 2 over 5 sin left parenthesis 180 t right parenthesis degree where t is the time in seconds after timing commenced.

Find the first time the seagull is 0.3 metres above its calm water position.
Give your answer to 2 decimal places.

How did you do?

9b
Sme Calculator2 marks

How many times in the first minute after timing commences is the seagull 0.3 metres above its calm water position?

How did you do?

Did this page help you?

1
Sme Calculator3 marks

Solve the equation 3 space sin 3 theta equals 4 space cos 3 theta in the interval 0 less or equal than theta less or equal than pi, giving your answers to 3 significant figures.

How did you do?

Did this page help you?

2
Sme Calculator5 marks

Solve the equation 6 cos squared 2 theta equals sin 2 theta plus 5 for negative 180 degree less or equal than theta less or equal than 180 degree , giving your answers to 1 decimal place where appropriate.

How did you do?

Did this page help you?

3
Sme Calculator3 marks

Given that the angle theta is reflex and that cos theta equals 1 third, find the exact value of tan space theta.

How did you do?

Did this page help you?

4
Sme Calculator5 marks

Solve the equation 2 sin squared 3 x equals 1 for negative pi over 2 less or equal than x less or equal than pi over 2.

How did you do?

Did this page help you?

5
Sme Calculator5 marks

Solve the equation 3 sin left parenthesis 2 x plus 30 degree right parenthesis equals tan left parenthesis 2 x plus 30 degree right parenthesis for negative 180 degree less or equal than x less or equal than 180 degree, giving your answers to 1 decimal place where appropriate.

How did you do?

Did this page help you?

6
Sme Calculator6 marks

For the triangle in the diagram, find exact values for sin space x comma space cos space x space and space tan space x.

Triangle diagram with sides 8 cm, 7 cm, and 12 cm. Angle x is at a vertex. Note says "Diagram not to scale."

How did you do?

Did this page help you?

7
Sme Calculator6 marks

Find all the values of x in the range 0 degree less or equal than x less or equal than 180 degree which satisfy the equation 6 tan cubed 2 x minus 7 tan squared 2 x minus tan 2 x plus 2 equals 0 comma giving your answers to 1 decimal place.

How did you do?

Did this page help you?

8a
Sme Calculator6 marks

Find all the solutions to the equation 2 space cos 2 theta equals 4 space sin 2 theta space space cos 2 theta in the interval 0 less or equal than theta less or equal than 2 pi giving your answers in radians as multiples of pi.

How did you do?

8b
Sme Calculator6 marks

Find all the solutions to the equation 3 space cos squared 4 x plus 13 space cos 4 x minus 10 equals 0 in the interval 0 less or equal than x less or equal than pi, giving your answers in radians to three significant figures.

How did you do?

Did this page help you?

9
Sme Calculator7 marks

A seagull sits on the surface of the sea and moves up and down as waves pass.

Its height, h metres, above its position in calm water is modelled by the function h equals 3 over 5 sin left parenthesis 90 t right parenthesis degree where t is the time in seconds after timing commences.

Find the amount of time the seagull is more than 0.5 metres above its calm water position in the first 20 seconds after timing commences.

Give your answer correct to 3 significant figures.

How did you do?

Did this page help you?