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

Exam code: 9MA0

4 hours43 questions
Easy
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Hard
Very Hard
13 marks

Solve the equation

sec space theta equals 1

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

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23 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.

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33 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.

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4a2 marks

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

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4b1 mark

Find the solution of the equation

arcsin space x equals pi over 4

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5
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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.

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63 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.

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7a2 marks

State the domain and range of the function

straight f open parentheses x close parentheses equals arccos space x

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7b1 mark

Solve the equation

Error converting from MathML to accessible text.

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8a2 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.

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8b
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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.

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9a2 marks

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

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9b2 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.

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103 marks

Solve the equation

cosec space 2 theta equals 2 

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

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11a2 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

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11b3 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

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14 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.

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23 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.

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3a4 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.

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3b
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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.

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3c
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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.

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4a2 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

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4b
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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.

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55 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.

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66 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.

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7a2 marks

State the domain and range of the function

straight f open parentheses x close parentheses equals arcsin space x

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7b3 marks

Solve the equation

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

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8a2 marks

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

cosec space theta equals 10

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8b2 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.

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8c
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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).

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9
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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.

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1a3 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.

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1b
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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

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2a3 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

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2b5 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

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35 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.

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45 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.

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5
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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.

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65 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.

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7a
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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.

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7b
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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.

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7c
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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.

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86 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.

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9a4 marks

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

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9b1 mark

State the solution to the equation

arccos space x equals arcsin space x

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10a
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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.

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10b3 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.

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10c
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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).

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114 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

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125 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.

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135 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.

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1a3 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

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1b
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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.

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25 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.

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3a4 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.

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3b4 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).

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4a
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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.

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4b
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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.

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5
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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.

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6a3 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

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6b4 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

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7a3 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

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7b2 marks

Explain why

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

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87 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.

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95 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.

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10a4 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.

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10b
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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.

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