International ASMathsEdexcelPure 2Exam QuestionsTrigonometryTrigonometric Equations

Trigonometric Equations (Edexcel International AS Maths: Pure 2): Exam Questions

Exam code: XMA01

3 hours36 questions
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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 with sides 12 cm, 13 cm, angle θ, and right angle marked. Note: The image is not to scale.

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1b
Sme Calculator3 marks

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

(i) sin space theta

(ii) cos space theta

(iii) tan space theta

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2
Sme Calculator2 marks

Show that

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

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3
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Solve the equation

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

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4a
Sme Calculator2 marks

Solve the equation x squared plus x minus 2 equals 0.

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4b
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Hence, or otherwise, solve the equation cos squared space x plus cos space x minus 2 equals 0 for 0 degree less or equal than x less or equal than 720 degree.

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5
Sme Calculator3 marks

Solve the equation tan space 2 theta equals 0.3 for negative 180 degree less or equal than theta less or equal than 180 degree, giving your answers to one decimal place.

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6a
Sme Calculator2 marks

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

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6b
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Solve the equation cos space 2 x equals 0.5 for 0 less or equal than x less or equal than 2 pi.

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7
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Solve the equation 2 left parenthesis 1 minus cos squared space theta right parenthesis equals 1 for negative pi less or equal than theta less or equal than pi.

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8
Sme Calculator4 marks

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

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9
Sme Calculator4 marks

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

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1a
Sme Calculator4 marks

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

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1b
Sme Calculator6 marks

Find all solutions to the equation 5 space sin space 3 x equals 1in the interval 0 less or equal than x less or equal than pi, giving your answer in radians to three significant figures.

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2a
Sme Calculator2 marks

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

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2b
Sme Calculator3 marks

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

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3
Sme Calculator3 marks

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

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4
Sme Calculator3 marks

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

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5
Sme Calculator5 marks

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

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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 space cos space theta and tan space theta, where theta is the smallest angle in the triangle.

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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.

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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

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8b
Sme Calculator5 marks

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

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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 functionh equals 1 half space sin space left parenthesis 180 t right parenthesis where tis 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

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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?

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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.

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1
Sme Calculator3 marks

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

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2
Sme Calculator5 marks

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

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3
Sme Calculator3 marks

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

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4
Sme Calculator5 marks

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

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5
Sme Calculator5 marks

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

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6
Sme Calculator6 marks

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

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

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7a
Sme Calculator4 marks

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

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7b
Sme Calculator5 marks

Find all the solutions to the equation 6 space sin squared space x plus 7 space sin space 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.

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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.

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8b
Sme Calculator6 marks

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

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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 functionh 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.

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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?

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1
Sme Calculator3 marks

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

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2
Sme Calculator5 marks

Solve the equation 6 space cos squared 2 theta equals sin space 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.

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3
Sme Calculator3 marks

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

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4
Sme Calculator5 marks

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

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5
Sme Calculator5 marks

Solve the equation3 space 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.

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6
Sme Calculator6 marks

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

Triangle diagram with side lengths 12 cm, 8 cm, and 7 cm; angle x marked inside; labelled "Diagram not to scale" on the right.

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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 space tan cubed 2 x minus 7 space tan squared 2 x minus tan 2 x plus 2 equals 0, giving your answers to 1 decimal place.

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8a
Sme Calculator6 marks

Find all the solutions to the equation

2 space cos 2 theta equals 4 space sin 2 theta 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.

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8b
Sme Calculator6 marks

Find all the solutions to the equation 3 cos squared 4 x plus 13 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.

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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.

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