Revision NotesExam QuestionsPast PapersMock Exams
A LevelMathsEdexcelPureExam QuestionsTrigonometryTrigonometric Equations

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

Exam code: 9MA0

3 hours30 questions
Easy
Medium
Hard
Very Hard
12 marks

Solve the equation

sin space 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?

Did this page help you?

2a2 marks

Solve the equation 

x squared plus x minus 2 equals 0

How did you do?

2b2 marks

Hence solve

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.

How did you do?

Did this page help you?

33 marks

Solve

cos space 2 x equals 1 half

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

How did you do?

Did this page help you?

4
Sme Calculator3 marks

Solve

tan space 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?

Did this page help you?

54 marks

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

1 minus cos squared space theta equals 1 half

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

How did you do?

Did this page help you?

64 marks

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

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

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

How did you do?

Did this page help you?

73 marks

Solve the equation

2 sin space 2 theta equals 1

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

How did you do?

Did this page help you?

1a2 marks

Express the equation

2 sin squared space x plus 3 cos space x equals 0

in the form

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

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

How did you do?

1b3 marks

Hence solve the equation

2 sin squared space x plus 3 cos space 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?

2
Sme Calculator3 marks

Solve the equation

2 sin space theta equals 3 cos space 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?

Did this page help you?

33 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

How did you do?

Did this page help you?

44 marks

Solve the equation

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

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

How did you do?

Did this page help you?

55 marks

Solve the equation

2 sin squared space theta equals 1 plus cos space theta

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

How did you do?

Did this page help you?

64 marks

Solve the equation

2 sin space x cos space x equals cos space x

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

How did you do?

Did this page help you?

7a3 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 halfsinopen 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?

7b3 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?

Did this page help you?

85 marks

Solve the equation

2 sin squared space 3 x equals 1

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

How did you do?

Did this page help you?

9a2 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
Sme Calculator5 marks

Hence solve the equation

tan cubed space x minus 4 tan squared space x plus tan space 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?

Did this page help you?

1a
Sme Calculator3 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?

1b2 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?

1c1 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?

Did this page help you?

26 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 .

How did you do?

Did this page help you?

3
Sme Calculator4 marks

Solve the equation 

3 sin space 3 theta equals 4 cos space 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?

Did this page help you?

4
Sme Calculator3 marks

Given that angle theta is obtuse where

sin space theta equals 3 over 4

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

How did you do?

Did this page help you?

54 marks

Solve the equation

2 tan space x minus sin space 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?

6a1 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?

6b8 marks

Hence solve

8 cos cubed space x minus 4 cos squared space x minus 6 cos space 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?

7
Sme Calculator7 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?

Did this page help you?

8
Sme Calculator3 marks

Given that angle theta is reflex where

cos space theta equals 1 third

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

How did you do?

Did this page help you?

1a3 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
Sme Calculator5 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?

Did this page help you?

2a
Sme Calculator5 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
Sme Calculator2 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
Sme Calculator2 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?

Did this page help you?

3a3 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
Sme Calculator4 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
Sme Calculator2 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?

Did this page help you?

46 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?

Did this page help you?

5
Sme Calculator6 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?

Did this page help you?

6
Sme Calculator8 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 space 2 x minus 7 tan squared space 2 x minus tan space 2 x plus 2 equals 0

giving your answers to 1 decimal place.

How did you do?

Did this page help you?