Trigonometric Equations (OCR AS Maths: Pure)

Exam Questions

2 hours32 questions
1a
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2 marks

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

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1b
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3 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
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2 marks

Show that

                                                                                                                    fraction numerator 1 minus cos to the power of 2 space end exponent x over denominator tan squared space x end fraction identical to cos to the power of 2 space end exponent x

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3
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3 marks

Solve the equation

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

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4a
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2 marks

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

4b
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2 marks

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 space x less or equal than 720 degree.

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5
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3 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
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2 marks

Sketch the graph of y equals cos space 2 x for  0 degree less or equal than space x less or equal than 360 degree.

6b
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2 marks

Solve the equation cos space 2 x equals 0.5 for  0 degree less or equal than space x less or equal than 360 degree.

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7
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4 marks

Solve the equation 2 open parentheses 1 minus cos to the power of 2 space end exponent theta close parentheses equals 1 for  negative 180 degree less or equal than theta less or equal than 180 degree.

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8
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4 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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1
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3 marks

Solve the equation  2 sin 2 theta equals 1 for  0 degree less or equal than theta less or equal than 360 degree.

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2a
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2 marks

Show that the equation  2 sin2 xplus 3 cos xequals 0  can be written in the form  a cosxplus b cos xplus c equals 0 , where a, b and c are integers to be found.

2b
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3 marks

Hence, or otherwise, solve the equation  2 spacesin2 x plus 3cos x equals 0  for  negative 180 degree less or equal than space x less or equal than 180 degree.

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3
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3 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
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5 marks

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

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5
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5 marks

Solve the equation 2 sin x cos xequalscos x for  negative 180 degree less or equal than space x less or equal than 180 degree.

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6
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6 marks

A right-angled triangle has hypotenuse 8 cm.  One of its other sides is 5 cm.

Find exact values for sin space theta, cos space theta and tan space theta, where theta is the smallest angle in the triangle.

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7a
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2 marks

Show that open parentheses x plus 1 close parentheses open parentheses x minus 2 close parentheses open parentheses x minus 3 close parentheses identical to x cubed minus 4 x squared plus x plus 6.

7b
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5 marks

Hence, or otherwise, solve the equation tan3 xnegative 4 tan2 xplustan xplus 6 equals 0  for  0 degree less or equal than space x less or equal than 360 degree, giving your answers to 1 decimal place where appropriate.

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8a
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2 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 halfsinopen parentheses 180 t close parentheses degree  where t is the time in seconds after timing commences.

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



8b
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1 mark

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

8c
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3 marks

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

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1
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3 marks

Solve the equation 2 sin thetaequals 3 cos theta  for 0 degree less or equal than theta less or equal than 360 degree, giving your answers to 1 decimal place.

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2
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5 marks

Solve the equation  2 sin2 thetaequalscos thetaplus 1 for  negative 180 degree less or equal than theta less or equal than 180 degree

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3
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3 marks

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

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4
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5 marks

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

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5
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5 marks

Solve the equation  2 spacetan x minussin x equals 0  for  negative 180 degree less or equal than space x less or equal than 180 degree.

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6
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6 marks

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

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

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7a
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1 mark

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

7b
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6 marks

Hence solve the equation 8 cos3 x minus 4 cos2 x minus 6 cos x plus 3 equals 0  for 0 degree less or equal than space x less or equal than 360 degree.

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8a
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4 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 5sinopen parentheses 180 t close parentheses 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.

8b
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2 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
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3 marks

Solve the equation  3 sin 3 thetaequals 4 cos 3 theta  for 0 degree less or equal than theta less or equal than 180 degree , giving your answers to 1 decimal place.

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2
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5 marks

Solve the equation 6 spacecos2 2 thetaequalssin 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
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3 marks

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

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4
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5 marks

Solve the equation   2 spacesin2 3 x equals 1  for  negative 90 degree less or equal than space x less or equal than 90 degree.

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5
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5 marks

Solve the equation  3 sinopen parentheses 2 x plus 30 degree close parenthesesequalstanopen parentheses 2 x plus 30 degree close parentheses   for  negative 180 degree less or equal than space x less or equal than 180 degree, giving your answers to 1 decimal place where appropriate.

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6
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6 marks

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

q6-5-3-trigonometric-equations-edexcel-a-level-pure-maths-veryhard

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7
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6 marks

Find all the values of x in the range 0 degree less or equal than space x less or equal than 180 degree  which satisfy the equation  6 spacetan3 2 x minus 7 spacetan2 2 x minustan 2 x plus 2 equals 0, giving your answers to 1 decimal place.

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8
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7 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 open parentheses 90 t close parentheses 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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