A LevelMaths: Pure 3CIEExam Questions3. Trigonometry3.3 Further Trigonometric Equations

Further Trigonometric Equations (CIE A Level Maths: Pure 3)

Exam Questions

2 hours32 questions
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1
Sme Calculator3 marks

Solve the equation sec theta equals 1 for  0 degree less or equal than theta less or equal than 360 degree.

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2
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Given that

tanopen parentheses A degree 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 degree less or equal than 180 degree.

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3
Sme Calculator3 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 comma space space space space space space space space space space space space space space space space space space space space minus straight pi less or equal than space x less or equal than straight pi

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4
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Use the identity

R space cos open parentheses A minus B close parentheses identical to R space cos space A space cos space B plus R space sin space A space sin space B

 to show that

8 space cos space theta plus 6 space sin space theta

can be written as

10 cos open parentheses theta minus straight alpha close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space spacewhere alpha equals 0.644 space to three significant figures.

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5a
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Show that the equation  cosec2 x equals 2 cosec x minus 1  can be written as

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

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5b
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Hence, or otherwise, solve the equation

cosec2 x equals 2 cosec x minus 1 comma space space space space space space space space space space space space space space space space space space space space space space space space minus 2 straight pi less or equal than space x less or equal than 2 straight pi

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6
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Solve the equation

cos 2 theta equals 1 half comma space space space space space space space space space space minus straight pi less or equal than theta less or equal than straight pi

State your answers as multiples of straight pi.

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7a
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Sketch the graph of  y equalssec x for  negative straight pi less or equal than space x less or equal than straight pi.

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7b
Sme Calculator2 marks
(i)
Add a line to your graph demonstrate how the equation

                                    sec x equals k          negative straight pi less or equal than space x less or equal than straight pi

where k is a constant could have no real solutions.

(ii)
For which values of k does this equation have no real solutions?

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8
Sme Calculator3 marks

Solve the equation

cot2 thetaminuscos theta cosec2 theta equals 0 comma space space space space space space space space space space space space space space space space space space space space space 0 less than theta less than 2 straight pi

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1
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Solve the equation cosec 2 theta equals 2 for 0 degree less or equal than theta less or equal than 180 degree.

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2
Sme Calculator3 marks

Given that

tanopen parentheses 3 A degree 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 degree less or equal than 120 degree.

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3
Sme Calculator4 marks

Solve the equation

fraction numerator sin space x over denominator sec space x end fraction equals 1 fourth comma space space space space space space space space space space space space minus straight pi less or equal than italic space x italic less or equal than straight pi

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4a
Sme Calculator4 marks

Use the identity


Rsinopen parentheses A plus B close parentheses identical to R spacecos B sin A plus Rsin B cos A

to show that

3 sin theta plus 4 cos theta

can be written as

5sinspace open parentheses theta plus straight alpha close parentheses comma space space space space space space space space space space space space where space space alpha equals tan to the power of negative 1 end exponent space open parentheses 4 over 3 close parentheses

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4b
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Hence, or otherwise, solve the equation 3 sin theta plus 4 cos theta equals 1 for  0 less or equal than theta less or equal than straight pi.
Give your answers to three significant figures.

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4c
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Write down the maximum value of 3 sin theta plus 4 spacecos theta and state the first positive value of theta for which it occurs.  Give your value of theta  to three significant figures.

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5a
Sme Calculator2 marks

Show that the equation  3 tan2 x equals 18 minus 2 sec x  can be written as

3 sec2 x plus 2sec x minus 21 equals 0

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5b
Sme Calculator4 marks

Hence, or otherwise, solve the equation

3 tan2 x equals 18 minus 2 sec x comma space space space space space space space space space space space space space space space space space space space space space space minus straight pi less or equal than space x less or equal than straight pi

Give your answers to three significant figures.

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6
Sme Calculator5 marks

Solve the equation

cos 2 theta equalscos theta minus 1 space space space space space space space space space space space minus straight pi less or equal than theta less or equal than straight pi

State your answers as multiples of straight pi.

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7a
Sme Calculator3 marks

Sketch the graph of y equals cot squared space theta for  negative 2 straight pi less or equal than theta less or equal than 2 straight pi.

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7b
Sme Calculator2 marks

By adding three lines to your graph demonstrate how the equation

cot2 theta equals straight k space space space space space space space space space space space minus 2 straight pi less or equal than theta less or equal than 2 straight pi

where k is a constant has either 0, 4 or 8 real solutions.

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8
Sme Calculator4 marks

Solve the equation

cot2 theta equalssec2 theta minus 1 comma space space space space space space space space space space space space 0 degree less or equal than theta less or equal than 360 degree

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1
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Solve the equation  sec2 2 x equals 1 plustan 2 x  for  0 degree less or equal than space x less or equal than 180 degree.

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2
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Given that

sinopen parentheses 2 A degree minus B degree 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  and 60 degree less than B degree less than A degree less than 300 degree

find the values of A and B.

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3
Sme Calculator4 marks

Solve the equation

fraction numerator cos space x over denominator cosec space x end fraction minus cot space x equals 0 comma space space space space space space space space minus 2 straight pi less or equal than italic space x less or equal than 2 straight pi

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4a
Sme Calculator3 marks

Show that 6 cos theta minus 8 sin theta can be written in the form  R cosopen parentheses theta plus straight alpha close parentheses, where R greater than 0 and alpha is an acute angle measured in radians.

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4b
Sme Calculator3 marks

Hence, or otherwise, solve the equation 3 cos theta minus 4 sin theta minus 2 equals 0,for 0 less or equal than space x less or equal than 2 straight pi
Give your answers to three significant figures.

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4c
Sme Calculator2 marks

Write down the minimum value of 6 cos theta minus 8 sin theta and the smallest positive value of theta for which it occurs.  Give your value of theta  to three significant figures.

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5
Sme Calculator4 marks

Solve the equation

2 cot2 x equals 8 minuscosec x comma space space space space space space space space minus straight pi less or equal than x less or equal than straight pi

giving your answers to three significant figures where appropriate.

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6
Sme Calculator5 marks

Solve the equation

8 spacecos4 theta minus 5 cos 2 theta minus 2 equals 0 space space space space space space space space space space 0 less or equal than theta less or equal than straight pi

State your answers as multiples of straight pi.

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7
Sme Calculator4 marks

Determine the values of the constant k for which the equation

                  cosec theta equals straight k comma space space space space space space minus straight pi less or equal than theta less or equal than 2 straight pi

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

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8
Sme Calculator4 marks

Solve the equation

cot2 theta equals 15 minus 6cosec theta comma space space space space space space space space space space space space minus 180 degree less or equal than theta less or equal than 180 degree

Give your answers to one decimal place where appropriate.

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1
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Solve the equation sin3 3 theta minussin 3 theta cos2 3 theta equals 0  for  0 degree less or equal than theta less than 180 degree.

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2
Sme Calculator4 marks

Given that

cosopen parentheses A degree minus B degree close parentheses equals negative fraction numerator square root of 3 over denominator 2 end fraction and  tanopen parentheses 1 half A degree minus B degree close parentheses equals square root of 3

and that

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

find the possible values of A and B.

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3a
Sme Calculator4 marks

Solve the equation 5 sin theta plus 2 cos theta equals 3,for  negative straight pi less or equal than theta less or equal than straight pi.
Give your answers to three significant figures.

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3b
Sme Calculator2 marks

Write down the maximum value of 5 sin theta plus 2 cos theta and the second positive value of theta  for which it occurs.  Give your value of  to three significant figures.

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4
Sme Calculator4 marks

Solve the equation

3 sec4 theta plus 16 equals 16 plus 16 spacetan2 theta comma space space space space space space space space space space space minus straight pi less or equal than theta less or equal than straight pi

giving your answers to three significant figures where appropriate.

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5
Sme Calculator5 marks

Solve the equation

cosec2 x minus 2 fraction numerator cosec space x over denominator sec space x end fraction equals 9 space space space space space space space space space space space space 0 less or equal than space x less or equal than 2 straight pi

Give your answers to three significant figures.

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6
Sme Calculator5 marks

Solve the equation

8sin4 2 theta equals 2 minus 5 cos 4 theta space space space space space space space space space minus straight pi over 2 less or equal than theta less or equal than straight pi over 2

State your answers as multiples of straight pi.

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7
Sme Calculator5 marks

The number of real solutions to the equation

open vertical bar sec space x minus 2 close vertical bar equals k comma space space space space space space space space space minus 2 straight pi less or equal than space x less or equal than 2 straight pi

is determined by the value of the constant k.

Find the number of real solutions for all values of k, given that k element of straight real numbers.

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8a
Sme Calculator4 marks

Given that x equals negative 2 is a root of x cubed plus 12 x squared plus 44 x plus 48, solve the equation by factorisation.

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8b
Sme Calculator4 marks

Solve the equation

sec space theta open parentheses sec squared space theta plus 44 close parentheses plus 12 open parentheses tan squared space theta plus 5 close parentheses equals 0 comma space space space space space space space space space space 0 degree less or equal than theta less or equal than 180 degree

Give your answers to one decimal place where appropriate.

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