ASMathsEdexcelPureExam QuestionsTrigonometryTrigonometric Equations

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

Exam code: 8MA0

2 hours32 questions
Easy
Medium
Hard
Very Hard
1a
Sme Calculator2 marks

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

q1a-5-3-trigonometric-equations-edexcel-a-level-pure-maths-easy

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1b
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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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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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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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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 space x less or equal than 720 degree.

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5
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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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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.

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6b
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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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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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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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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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Show that the equation  2 sin2 xplus 3 cos xequals 0  can be written in the form  a cos2 xplus b cos xplus c equals 0 , where a, b and c are integers to be found.

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2b
Sme Calculator3 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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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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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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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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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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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.

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7b
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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
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 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.

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8b
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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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8c
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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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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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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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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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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 space x less or equal than 180 degree .

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5
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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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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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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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7b
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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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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.

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8b
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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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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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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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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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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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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
Sme Calculator6 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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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
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 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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