A LevelMathsEdexcelPureExam QuestionsTrigonometryFurther Trigonometric Equations

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

Exam code: 9MA0

4 hours43 questions
Easy
Medium
Hard
Very Hard
1
3 marks

Solve the equation

sec space theta equals 1

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

How did you do?

2
3 marks

Given that

tan open parentheses A minus 30 degree close parentheses equals fraction numerator square root of 3 over denominator 3 end fraction

find the values of A such that  negative 180 degree less or equal than A less or equal than 180 degree.

How did you do?

3
3 marks

Solve the equation

fraction numerator 1 over denominator sec space x end fraction equals fraction numerator square root of 2 over denominator 2 end fraction

for negative pi less or equal than space x less or equal than pi.

How did you do?

4a
2 marks

Sketch the curve y equals arcsin space x, where x is in radians.

How did you do?

4b
1 mark

Find the solution of the equation

arcsin space x equals pi over 4

How did you do?

5
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4 marks

Show that

8 cos space theta plus 6 sin space theta

can be written as

10 cos open parentheses theta minus straight alpha close parentheses

where alpha equals 0.644 spaceradians, to 3 significant figures.

How did you do?

6
3 marks

Solve the equation

cos space 2 theta equals 1 half

for negative pi less or equal than theta less or equal than pi, giving your answers in an exact form.

How did you do?

7a
2 marks

State the domain and range of the function

straight f open parentheses x close parentheses equals arccos space x

How did you do?

7b
1 mark

Solve the equation

Error converting from MathML to accessible text.

How did you do?

8a
2 marks

Use the small angle approximation tan space theta almost equal to theta to find an estimate of the solution to the equation

4 cot space theta minus 2 equals 3

in the interval 0 less than theta less than pi over 2.

How did you do?

8b
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3 marks

Solve

4 cot space theta minus 2 equals 3

for 0 less than theta less than pi over 2, giving your answer to 3 significant figures.

How did you do?

9a
2 marks

Sketch the curve y equals sec space x for negative pi less or equal than space x less or equal than pi.

How did you do?

9b
2 marks

(i) By adding a suitable straight line to your sketch in part (a), show that the equation

sec space x equals 1 half

where negative pi less or equal than space x less or equal than pi has no real solutions.

(ii) Find the possible values of k for which the equation

sec space x equals k

has no real solutions.

How did you do?

10
3 marks

Solve the equation

cosec space 2 theta equals 2 

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

How did you do?

11a
2 marks

Show that the equation

cosec squared x equals 2 cosec space x minus 1

 can be written as

open parentheses cosec space x minus 1 close parentheses squared equals 0

How did you do?

11b
3 marks

Hence solve for negative 2 pi less or equal than x less or equal than 2 pi

cosec squared x equals 2 cosec space x minus 1

How did you do?

1
4 marks

Solve the equation

fraction numerator sin space x over denominator sec space x end fraction equals 1 fourth

for negative pi less or equal than italic space x italic less or equal than pi.

How did you do?

2
3 marks

Given that

tan open parentheses 3 A minus 30 degree close parentheses equals negative fraction numerator square root of 3 over denominator 3 end fraction

find the values of A such that  negative 120 degree less or equal than A less or equal than 120 degree.

How did you do?

3a
4 marks

Write

3 sin space theta plus 4 cos space theta

in the form

R sin open parentheses theta plus straight alpha close parentheses

where R greater than 0 and 0 less than alpha less than pi over 2.

Give alpha in the form tan to the power of negative 1 end exponent open parentheses p over q close parentheses where p and q are integers to be found.

How did you do?

3b
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3 marks

Hence solve the equation

3 sin space theta plus 4 cos space theta equals 1

for  0 less or equal than theta less or equal than pi.

Give your answers to 3 significant figures.

How did you do?

3c
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3 marks

State the maximum value of

3 sin space theta plus 4 cos space theta

and find the smallest positive value of theta for which this maximum value occurs,  to 3 significant figures.

How did you do?

4a
2 marks

Show that the equation 

3 tan squared x equals 18 minus 2 sec space x

can be written in the form

3 sec squared x plus 2 sec space x minus 21 equals 0

How did you do?

4b
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4 marks

Hence solve the equation

3 tan squared x equals 18 minus 2 sec space x

for negative pi less or equal than space x less or equal than pi, giving your answers to 3 significant figures.

How did you do?

5
5 marks

Solve the equation

cos space 2 theta equals cos space theta minus 1

for negative pi less or equal than space theta less or equal than pi, giving your answers in an exact form.

How did you do?

6
6 marks

Solve the equation

2 cot squared x equals 8 minus cosec space x

for negative pi less or equal than x less or equal than pi.

Give your answers to 3 significant figures, where necessary.

How did you do?

7a
2 marks

State the domain and range of the function

straight f open parentheses x close parentheses equals arcsin space x

How did you do?

7b
3 marks

Solve the equation

open square brackets straight f open parentheses x close parentheses close square brackets squared equals pi squared over 16

How did you do?

8a
2 marks

Use a small angle approximation to estimate the solution to the equation

cosec space theta equals 10

How did you do?

8b
2 marks

Find the exact solution to

cosec space theta equals 10

for 0 less than theta less than straight pi over 2 in the form

arcsin open parentheses 1 over k close parentheses

where k is an integer to be found.

How did you do?

8c
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2 marks

Find, to 2 significant figures, the percentage error in using the approximate solution from part (a) instead of the the exact solution from part (b).

How did you do?

9
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6 marks

Solve the equation

cot squared theta equals 15 minus 6 cosec space theta

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

Give your answers to one decimal place, where necessary.

How did you do?

1a
3 marks

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

Solutions relying entirely on calculator technology are not acceptable.

Show that the equation

2 tan theta space open parentheses 8 cos theta plus 23 sin squared theta close parentheses equals 8 sin open parentheses 2 theta close parentheses space open parentheses 1 plus tan squared theta close parentheses

may be written as

sin open parentheses 2 theta close parentheses space open parentheses A cos squared theta plus B cos theta plus C close parentheses equals 0

where A, B and C are constants to be found.

How did you do?

1b
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4 marks

Hence solve for 360 degree less or equal than x less or equal than 540 degree

2 tan x space open parentheses 8 cos x plus 23 sin squared x close parentheses equals 8 sin open parentheses 2 x close parentheses space open parentheses 1 plus tan squared x close parentheses space space space space space space space space space x element of straight real numbers space space space space x not equal to 450 degree

How did you do?

2a
3 marks

Show that

cosec space theta minus sin space theta identical to cos space theta space co t space theta space space space space space space space space theta not equal to open parentheses 180 n close parentheses degree space space space n element of straight integer numbers

How did you do?

2b
5 marks

Hence, or otherwise, solve for 0 less than x less than 180 degree

cosec x minus sin x equals cos x cross times cot open parentheses 3 x minus 50 degree close parentheses

How did you do?

3
5 marks

Solve the equation

cot squared theta minus cos space theta space cosec squared theta equals 0

for 0 less than theta less than 2 pi.

How did you do?

4
5 marks

Solve the equation

sec squared open parentheses 2 x close parentheses equals 1 plus tan open parentheses 2 x close parentheses

for  0 degree less or equal than space x less or equal than 180 degree.

How did you do?

5
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4 marks

Given that

sin open parentheses 2 A minus B close parentheses equals fraction numerator square root of 6 plus square root of 2 over denominator 4 end fraction

and that

3 A equals 4 B 

where 60 degree less than B less than A less than 300 degree, find the values of A and B.

How did you do?

6
5 marks

Solve the equation

fraction numerator cos space x over denominator cosec space x end fraction minus cot space x equals 0

for negative 2 pi less or equal than italic space x less or equal than 2 pi.

How did you do?

7a
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4 marks

Write

6 cos space theta minus 8 sin space theta

in the form

R cos open parentheses theta plus straight alpha close parentheses

where R greater than 0 and 0 less than alpha less than pi over 2, giving alpha to 3 significant figures.

How did you do?

7b
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4 marks

Solve the equation

3 cos space theta minus 4 sin space theta minus 2 equals 0

for  0 less or equal than theta less or equal than 2 pi.

Give your answers to 3 significant figures.

How did you do?

7c
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3 marks

State the minimum value of

6 cos space theta minus 8 sin space theta

and find the smallest positive value of theta for which this minimum value occurs,  to 3 significant figures.

How did you do?

8
6 marks

Solve the equation

8 cos to the power of 4 theta minus 5 cos open parentheses 2 theta close parentheses minus 2 equals 0

for 0 less or equal than theta less or equal than pi, giving your answers in an exact form.

How did you do?

9a
4 marks

On the same axes, sketch the curves y equals arcsin space x and  y equals arccos space x, where x is in radians.

How did you do?

9b
1 mark

State the solution to the equation

arccos space x equals arcsin space x

How did you do?

10a
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3 marks

Use a small angle approximation to estimate the positive solution to the equation

sec space 2 theta equals 1.05

Give your answer to 6 decimal places.

How did you do?

10b
3 marks

Find the exact solution to

sec space 2 theta equals 1.05

for 0 less than theta less than pi over 2 in the form

1 over p arccos open parentheses q over r close parentheses

where p, q and r are integers to be found.

How did you do?

10c
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2 marks

Find, to 2 significant figures, the percentage error in using the approximate solution from part (a) instead of the the exact solution from part (b).

How did you do?

11
4 marks

Find the possible values of the constant k for which the equation

cosec space theta equals k

for negative pi less or equal than theta less or equal than 2 pi has

(i) no real solutions
(ii) 1 real solution
(iii) 2 real solutions
(iv) 4 real solutions

How did you do?

12
5 marks

Solve the equation

cot squared theta equals sec squared theta minus 1

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

How did you do?

13
5 marks

Solve the equation

sin cubed open parentheses 3 theta close parentheses minus sin open parentheses 3 theta close parentheses cos squared open parentheses 3 theta close parentheses equals 0

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

How did you do?

1a
3 marks

Prove that

1 minus cos 2 theta identical to tan theta sin 2 theta comma space space space space space space theta not equal to fraction numerator open parentheses 2 n plus 1 close parentheses pi over denominator 2 end fraction comma space space space space n element of straight integer numbers

How did you do?

1b
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6 marks

Hence solve, for negative pi over 2 less than x less than pi over 2, the equation

open parentheses sec squared x minus 5 close parentheses open parentheses 1 minus cos 2 x close parentheses equals 3 tan squared x sin 2 x

Give any non-exact answer to 3 decimal places where appropriate.

How did you do?

2
5 marks

Given that

  • cos open parentheses A minus B close parentheses equals negative fraction numerator square root of 3 over denominator 2 end fraction

  • tan open parentheses 1 half A minus B close parentheses equals square root of 3

  • 0 less or equal than 2 B less than A less or equal than 360 degree

find the possible values of A and B.

How did you do?

3a
4 marks

Sketch the curve y equals pi plus arctan open parentheses 2 x minus 1 close parentheses for x element of straight real numbers.

Label clearly the coordinates of the point at which the curve meets the y-axis and state the equations of any asymptotes.

How did you do?

3b
4 marks

A different curve is given by y equals pi plus arctan open parentheses 1 minus 2 x close parentheses for x element of straight real numbers.

Find the coordinates of the point of intersection of this curve with

(i) the y-axis,

(ii) the curve in part (a).

How did you do?

4a
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7 marks

Solve the equation

5 sin space theta plus 2 cos space theta equals 3

for negative pi less or equal than theta less or equal than pi.

Give your answers to 3 significant figures.

How did you do?

4b
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4 marks

State the maximum value of

5 sin space theta plus 2 cos space theta

and find the second smallest positive value of theta for which this maximum value occurs,  to 3 significant figures.

How did you do?

5
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6 marks

Solve the equation

cosec squared x minus fraction numerator 2 cosec space x over denominator sec space x end fraction equals 9

for 0 less or equal than space x less or equal than 2 pi.

Give your answers to 3 significant figures.

How did you do?

6a
3 marks

Find the domain and range of the function

straight f open parentheses x close parentheses equals cos open parentheses arcsin space x close parentheses

How did you do?

6b
4 marks

Solve the equation

2 open square brackets straight f open parentheses x close parentheses close square brackets squared minus 3 straight f open parentheses x close parentheses plus 1 equals 0

How did you do?

7a
3 marks

Find the value of

(i) arcsin open parentheses sin fraction numerator 7 pi over denominator 6 end fraction close parentheses

(ii) arccos open parentheses cos open parentheses negative fraction numerator 2 pi over denominator 3 end fraction close parentheses close parentheses

How did you do?

7b
2 marks

Explain why

arctan open parentheses tan space pi close parentheses not equal to pi

How did you do?

8
7 marks

Solve the equation

8 sin to the power of 4 open parentheses 2 theta close parentheses equals 2 minus 5 cos open parentheses 4 theta close parentheses

for negative pi over 2 less or equal than theta less or equal than pi over 2.

Give your answers in an exact form.

How did you do?

9
5 marks

The number of real solutions, n, to the equation

open vertical bar sec space x minus 2 close vertical bar equals k

where negative 2 pi less or equal than space x less or equal than 2 pi is determined by the value of the constant k, where k element of straight real numbers.

Find the possible values of n and state the corresponding range of values of k for each n.

How did you do?

10a
4 marks

Given that x equals negative 2 is a solution to the cubic equation

x cubed plus 12 x squared plus 44 x plus 48 equals 0

solve the equation by factorisation.

How did you do?

10b
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5 marks

Solve

sec space theta open parentheses sec squared theta plus 44 close parentheses plus 12 open parentheses tan squared theta plus 5 close parentheses equals 0

where 0 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?