A LevelMathsEdexcelPureExam QuestionsTrigonometryTrigonometric Equations

Trigonometric Equations (Edexcel A Level Maths: Pure): Exam Questions

Exam code: 9MA0

3 hours30 questions
Easy
Medium
Hard
Very Hard
1
2 marks

Solve the equation

sin x equals 1 half

in the interval 0 degree less or equal than space x less or equal than 360 degree.

How did you do?

2a
2 marks

Solve the equation 

x squared plus x minus 2 equals 0

How did you do?

2b
2 marks

Hence solve

cos squared x plus cos x minus 2 equals 0

for 0 degree less or equal than space x less or equal than 720 degree.

How did you do?

3
3 marks

Solve

cos 2 x equals 1 half

for 0 degree less or equal than x less or equal than 360 degree.

How did you do?

4
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3 marks

Solve

tan 2 theta equals 3 over 10

for negative 180 degree less or equal than theta less or equal than 180 degree, giving your answers to 1 decimal place.

How did you do?

5
4 marks

Use the identity 1 minus cos squared theta identical to sin squared theta to solve the equation

1 minus cos squared theta equals 1 half

for negative 180 degree less or equal than theta less or equal than 180 degree.

How did you do?

6
3 marks

Use the identity 1 minus sin squared theta identical to cos squared theta to solve the equation

4 open parentheses 1 minus sin squared theta close parentheses equals 3

for 0 degree less or equal than theta less or equal than 180 degree.

How did you do?

7
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.

How did you do?

1a
2 marks

Express the equation

2 sin squared x plus 3 cos x equals 0

in the form

a cos squared x plus b cos x plus c equals 0

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

How did you do?

1b
3 marks

Hence 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?

2
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3 marks

Solve the equation

2 sin theta equals 3 cos theta

for 0 degree less or equal than theta less or equal than 360 degree.

Give your answers to 1 decimal place.

How did you do?

3
3 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?

4
4 marks

Solve the equation

2 sin x equals fraction numerator 1 over denominator sin x end fraction

for 0 degree less or equal than x less or equal than 360 degree.

How did you do?

5
5 marks

Solve the equation

2 sin squared theta equals 1 plus cos theta

for negative 180 degree less or equal than theta less or equal than 180 degree.

How did you do?

6
4 marks

Solve the equation

2 sin x cos x equals cos x

for negative 180 degree less or equal than x less or equal than 180 degree.

How did you do?

7a
3 marks

A seagull sits on the surface of the sea, moving up and down with the waves.

Its height, h metres, above sea level in calm water is modelled by

h equals 1 half sin open parentheses 180 t close parentheses degree

where t is the time in seconds after first being observed.

Sketch the 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.

How did you do?

7b
3 marks

Find the time at which the seagull is first observed to be 0.25 m above sea level in calm water and moving downwards.

Give your answer to 3 significant figures.

How did you do?

8
5 marks

Solve the equation

2 sin squared 3 x equals 1

for negative 90 degree less or equal than x less or equal than 90 degree.

How did you do?

9a
2 marks

Express

open parentheses x plus 1 close parentheses open parentheses x minus 2 close parentheses open parentheses x minus 3 close parentheses

in the form

a x cubed plus b x squared plus c x plus d

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

How did you do?

9b
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5 marks

Hence 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.

Give your answers to 1 decimal place where necessary.

How did you do?

1a
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3 marks

straight f open parentheses x close parentheses equals negative 3 x cubed plus 8 x squared minus 9 x plus 10 comma space space space space space x element of straight real numbers

(i) Calculate straight f open parentheses 2 close parentheses

(ii) Write straight f open parentheses x close parentheses as a product of two algebraic factors.

How did you do?

1b
2 marks

Using the answer to (a)(ii), prove that there are exactly two real solutions to the equation

negative 3 y to the power of 6 plus 8 y to the power of 4 minus 9 y squared plus 10 equals 0

How did you do?

1c
1 mark

Deduce the number of real solutions, for 7 pi less or equal than theta less than 10 pi, to the equation

3 tan cubed theta minus 8 tan squared theta plus 9 tan theta minus 10 equals 0

How did you do?

2
6 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?

3
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4 marks

Solve the equation 

3 sin 3 theta equals 4 cos 3 theta

for 0 degree less or equal than theta less or equal than 180 degree.

Give your answers to 1 decimal place.

How did you do?

4
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3 marks

Given that angle theta is obtuse where

sin theta equals 3 over 4

use a non-calculator method to find the exact value of cos theta.

How did you do?

5
4 marks

Solve the equation

2 tan x minus sin x equals 0

for negative 180 degree less or equal than x less or equal than 180 degree.

How did you do?

6a
1 mark

Show that x equals 1 half is a solution to the equation

8 x cubed minus 4 x squared minus 6 x plus 3 equals 0

How did you do?

6b
8 marks

Hence solve

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?

7
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7 marks

Solve the equation

6 cos squared open parentheses 2 theta close parentheses equals 5 plus sin open parentheses 2 theta close parentheses

for negative 180 degree less or equal than theta less or equal than 180 degree.

Give your answers to 1 decimal place where necessary.

How did you do?

8
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3 marks

Given that angle theta is reflex where

cos theta equals 1 third

use a non-calculator method to find the exact value of tan theta.

How did you do?

1a
3 marks

Show that

table row cell fraction numerator 1 over denominator cos theta end fraction plus tan theta identical to fraction numerator cos theta over denominator 1 minus sin theta end fraction end cell blank cell theta not equal to open parentheses 2 n plus 1 close parentheses 90 degree end cell blank cell n element of straight integer numbers end cell end table

How did you do?

1b
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5 marks

Given that cos space 2 x not equal to 0

solve for 0 less than x less than 90 degree

fraction numerator 1 over denominator cos space 2 x end fraction plus tan space 2 x equals 3 cos space 2 x

giving your answers to one decimal place.

How did you do?

2a
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5 marks

In this question you should show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Solve, for 0 less than theta less or equal than 450 degree, the equation

5 cos squared theta equals 6 sin theta

giving your answers to one decimal place.

How did you do?

2b
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2 marks

A student’s attempt to solve the question

“Solve, for – 90 degree less than x less than 90 degree, the equation space 3 tan x – 5 sin x equals 0”

is set out below.

3 tan x – 5 sin x equals 0

3 fraction numerator sin x over denominator cos x end fraction minus 5 sin x equals 0

3 sin x – 5 sin x cos x equals 0

3 – 5 cos x equals 0

cos x equals 3 over 5

x equals 53.1 degree

Identify two errors or omissions made by this student, giving a brief explanation of each.

How did you do?

2c
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2 marks

The first four positive solutions, in order of size, of the equation

cos open parentheses 5 alpha plus 40 degree close parentheses equals 3 over 5

are alpha subscript 1 , alpha subscript 2, alpha subscript 3 and alpha subscript 4

Find, to the nearest degree, the value of alpha subscript 4

How did you do?

3a
3 marks

In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.

Show that the equation

4 tan x equals 5 cos x

can be written as

5 sin squared x plus 4 sin x minus 5 equals 0

How did you do?

3b
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4 marks

Hence solve, for 0 less than x less or equal than 360 degree

4 tan x equals 5 cos x

giving your answers to one decimal place.

How did you do?

3c
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2 marks

Hence find the number of solutions of the equation

4 tan 3 x equals 5 cos 3 x

in the interval 0 less than x less or equal than 1800 degree, explaining briefly the reason for your answer.

How did you do?

4
6 marks

Solve the equation

3 sin open parentheses 2 x plus 30 degree close parentheses equals tan open parentheses 2 x plus 30 degree close parentheses

for negative 180 degree less or equal than x less or equal than 180 degree.

Give your answers to 1 decimal place where necessary.

How did you do?

5
6 marks

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

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

How did you do?

6
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8 marks

Find all the values of x in the interval 0 degree less or equal than space x less or equal than 180 degree  which satisfy

6 tan cubed 2 x minus 7 tan squared 2 x minus tan 2 x plus 2 equals 0

giving your answers to 1 decimal place.

How did you do?