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Reciprocal & Inverse Trigonometric Functions (Edexcel A Level Maths: Pure): Exam Questions

3 hours35 questions
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12 marks

Sketch the graph of y equals cosec space x for  negative 180 degree less or equal than space x less or equal than 180 degree.

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2a2 marks

Write down the domain and range of the function arc cos space theta.

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2b2 marks

Sketch the graph of y equalsarccos theta.

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3
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Solve the equation cot space x equals 3 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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43 marks

Sketch the graph of y equals sec space theta for  negative pi less or equal than theta less or equal than pi.

Label any points of intersection with the coordinate axes and state the equations of any asymptotes.

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52 marks

Starting with the identity

sin squared x plus cos squared x identical to 1

    show that

(i) 1 plus cot squared x identical to cosec squared x

(ii) tan squared x plus 1 identical to sec squared x

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6a2 marks

Write down the domain and range of the function arcsin space theta.

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6b
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Sketch the graph of y equals arc sin theta.

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7a2 marks

Write down the domain and range of the function arc tan theta.

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7b2 marks

Sketch the graph of y equals arc tan theta.

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82 marks

Show that

sec squared theta space sin space theta identical to tan space theta space sec space theta

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9a2 marks

Show that

sec space theta space cot space theta identical to cosec space theta

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9b3 marks

Hence solve, in the range 0 less or equal than theta less or equal than italic 2 pi, the equation

sec space theta space cot space theta equals negative 2

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13 marks

Show that

cosec space theta minus sin space theta identical to cos space theta cot 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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24 marks

Solve the equation

sec space theta space tan space theta minus sec space theta equals 0

for 0 less or equal than space x less or equal than 2 straight pi, giving your answers in exact form.

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33 marks

Show that

cot space x space cosec space x space sec space x identical to 1 plus cot squared x

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4
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Solve the equation

cosec squared x minus 2 cosec space x minus 8 equals 0

for 0 degree less or equal than space x less or equal than 360 degree, giving your answers to 1 decimal place where appropriate.

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5a2 marks

Show that the equation

3 minus sec space theta equals fraction numerator 2 over denominator sec space theta end fraction

may be written in the form

 open parentheses sec space theta minus 2 close parenthesesopen parentheses sec space theta minus 1 close parentheses equals 0

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5b4 marks

Hence solve

3 minus sec space theta equals fraction numerator 2 over denominator sec space theta end fraction

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

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6a3 marks

Show that the equation

tan squared x equals 6 sec space x minus 10

may be written in the form

 open parentheses sec space x minus 3 close parentheses squared equals 0

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6b
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Hence solve for 0 less or equal than space x less or equal than 2 pi

tan squared x equals 6 sec space x minus 10

giving your answers correct to 3 significant figures.

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7a3 marks

Sketch the graph of y equals 2 space sec space 2 x for negative pi less or equal than space x less or equal than pi.

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7b1 mark

Use the sketch in part (a) to find the number of solutions to the equation

2 space sec space 2 x equals 4

in the interval  negative pi less or equal than space x less or equal than pi.

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8
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(i) For negative 2 pi less or equal than theta less or equal than 2 pi, sketch the curve y equals 3 plus 2 cosec space theta, showing the coordinates of any maximum and minimum points.

(ii) Hence, find the number of solutions to the equation

3 plus 2 cosec space theta equals 1 half

in the interval  negative 2 pi less or equal than theta less or equal than 2 pi.

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9a1 mark

Write tan space theta space cosec space theta as a single trigonometric function.

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9b3 marks

Hence solve for negative pi less than theta less or equal than pi

tan space theta space cosec space theta equals negative fraction numerator 2 square root of 3 over denominator 3 end fraction

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1a3 marks

The function straight f is defined by

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

for negative 1 less or equal than space x less or equal than 1, and the function straight g is defined by

straight g open parentheses x close parentheses equals straight f open parentheses 3 x close parentheses

Sketch the graph of y equals straight f open parentheses x close parentheses  and state the range of  straight f.

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1b3 marks

Sketch the graph of y equals straight g open parentheses x close parentheses and state the domain of straight g.

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1c2 marks

Find straight g to the power of negative 1 end exponent open parentheses x close parentheses and state its domain.

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2a3 marks

Show that the equation

2 cot squared x equals 1 minus 5 cosec space x

can be written in the form

open parentheses 2 space cosec space x minus 1 close parentheses open parentheses cosec space x plus 3 close parentheses equals 0

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2b
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Hence solve, for 0 less or equal than space x less or equal than 2 pi, the equation

2 cot squared x equals 1 minus 5 cosec space x

giving your answers correct to 3 significant figures.

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36 marks

Solve, for 0 less or equal than theta less or equal than 2 pi, the equation

fraction numerator 2 over denominator cosec space theta end fraction minus cosec space theta equals 1

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45 marks

Given that x satisfies the equation arccos space x equals k , where   0 less than k less than pi over 2.

(i) state the range of possible values of x,

(ii) find both sin space k and tan space k in terms of x.

 ,

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5a3 marks

Show that the equation

sec space x space cosec space x minus 5 equals cosec space 2 x

may be written in the form

cosec space 2 x equals 5

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5b
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Hence solve for 0 less or equal than space x less or equal than 2 pi

sec space x space cosec space x minus 5 equals cosec space 2 x

giving your answers correct to 3 significant figures.

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65 marks

Given that x satisfies the equation  arc sin x equals k, where  negative pi over 2 less than k less than 0  ,

(i) state the range of possible values of x,

(ii) find both cos space k and tan space k in terms of x.

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7a3 marks

The function straight f is defined by

straight f open parentheses x close parentheses equals arctan space x

where  x element of straight real numbers , and the function straight g is defined by

straight g open parentheses x close parentheses equals 2 over straight pi straight f open parentheses x close parentheses minus 1 

Sketch the graph of y equals straight f open parentheses x close parentheses and state the range of straight f.

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7b3 marks

Sketch the graph of y equals straight g open parentheses x close parenthesesand state the range of straight g.

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7c2 marks

Find straight g to the power of negative 1 end exponent open parentheses x close parentheses and state its domain.

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85 marks

(i) Sketch, for negative 2 pi less or equal than theta less or equal than 2 pi, the curve y equals negative 5 plus 1 half sec space theta.

Label clearly the coordinates of any maximum and minimum points.

(ii) Hence find the range of values of k for which the equation   

negative 5 plus 1 half sec space theta equals k

has no solutions.

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1
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Solve for negative pi less than space x italic less or equal than pi

open parentheses cosec space x italic minus sec space x close parentheses open parentheses fraction numerator 1 over denominator sec space x end fraction plus fraction numerator 1 over denominator cosec space x end fraction close parentheses equals cot space 2 x plus 3

Give your answers correct to 3 significant figures.

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2
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Solve

sec space x space cosec space x minus 75 equals 5 cosec space 2 x

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

Give your answers correct to 3 significant figures.

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35 marks

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

fraction numerator sec space theta space cot space theta over denominator cosec space theta space tan space theta end fraction equals negative square root of 3

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4
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Solve for 0 less or equal than theta less or equal than 2 pi

6 sec space theta plus fraction numerator 2 square root of 3 over denominator sec space theta end fraction equals negative 3 minus 4 square root of 3

Leaving your answers exact.

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56 marks

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

3 space cot squared space x minus 4 square root of 3 equals open parentheses 6 minus 2 square root of 3 close parentheses cosec space x minus 3

giving your answers in an exact form.

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65 marks

Given that x satisfies the equation  arctan space x equals k where  negative pi over 2 less or equal than space k less or equal than 0

(i) state the range of possible values of x,

(ii) find both sin space k and cos space k in terms of x.

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7a4 marks

Sketch, for negative 2 pi less or equal than theta less or equal than pi, the curve y equals 2 plus 3 space sec open parentheses theta plus straight pi over 2 close parentheses .

Label clearly the coordinates of any maximum and minimum points.

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7b2 marks

Find the maximum and minimum values of

fraction numerator 1 over denominator 2 plus 3 space sec space open parentheses theta plus straight pi over 2 close parentheses end fraction

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8a5 marks

The function straight f is defined by

straight f open parentheses x close parentheses equals arcsin space x

for  negative 1 less or equal than space x less or equal than 1 , and the function straight g is defined by 

straight g open parentheses x close parentheses equals fraction numerator 4 straight f open parentheses x over 3 close parentheses over denominator straight pi end fraction plus 2

Sketch the graph of y equals straight g open parentheses x close parenthesesand state the domain and range of straight g.

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8b3 marks

Find straight g to the power of negative 1 end exponent open parentheses x close parentheses and state its domain.

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8c2 marks

Over the same domain as straight g, the function straight h is defined by

straight h open parentheses x close parentheses equals p arccos open parentheses straight q x close parentheses

Given that straight h open parentheses x close parentheses equals negative straight g open parentheses x close parentheses for all x in the domain, find the values of p and q.

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9a4 marks

Show that for 0 less or equal than space x less or equal than 1

arc sin x equals arc cossquare root of 1 minus x squared end root

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9b2 marks

Explain why the relationship in part (a) is not true for negative 1 less or equal than space x less than 0.

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