A LevelMaths: PureOCRExam Questions5. Trigonometry5.5 Reciprocal & Inverse Trigonometric Functions (A Level only)

Reciprocal & Inverse Trigonometric Functions (A Level only) (OCR A Level Maths: Pure)

Exam Questions

3 hours36 questions
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1
Sme Calculator2 marks

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

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2a
Sme Calculator2 marks

Write down the domain and range for the function arccos theta.

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2b
Sme Calculator2 marks

Hence sketch the graph of y equalsarccos theta.

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3
Sme Calculator3 marks

Solve the equation cot x equals 3, for  negative straight pi less or equal than space x less or equal than straight pi, giving your answers to three significant figures.

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4
Sme Calculator4 marks

Sketch the graph of y equalssec theta, for  negative straight pi less or equal than theta less or equal than straight pi.

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

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5
Sme Calculator4 marks

Starting with the identity

         sin2 x pluscos2 x identical to 1

show that

(i)

1 pluscot2 x identical tocosec2 x

(ii)
tan2 x plus 1 identical tosec2 x

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6
Sme Calculator3 marks

Show that

      sec2 theta sin theta identical totan theta sec theta

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7a
Sme Calculator2 marks

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

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7b
Sme Calculator2 marks

Hence sketch the graph of y equals arc sin theta.

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8
Sme Calculator3 marks

Solve the equation cosec2 x minus 2 cosec x minus 8 equals 0  for 0 degree less or equal than space x less or equal than 360 degree, giving your answers to one decimal place where appropriate.

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9
Sme Calculator3 marks

Show that

         cot x cosec x sec x identical to 1 pluscot2 x

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10
Sme Calculator4 marks

Solve the equation sec theta tan theta minussec 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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11a
Sme Calculator2 marks

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

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11b
Sme Calculator2 marks

Hence sketch the graph of y equals arc tan theta.

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12a
Sme Calculator2 marks

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

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12b
Sme Calculator2 marks

Draw a suitable line on your graph to show that the equation 2 sec 2 x equals 4  has four solutions in the range  negative straight pi less or equal than space x less or equal than straight pi.

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1a
Sme Calculator2 marks

Use the definitions of the secant, cosecant and cotangent functions to show that

sec theta cot theta identical tocosec theta.  

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1b
Sme Calculator3 marks

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

sec theta cot theta equals negative 2

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2a
Sme Calculator2 marks

Show that the equation

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

can be rewritten 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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2b
Sme Calculator4 marks

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

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

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3a
Sme Calculator3 marks

Using the double angle formula sin 2 A identical to 2 sin A cos A, show that the equation

sec x cosec x minus 5 equalscosec 2 x

can be rewritten in the form

coesec 2 x equals 5.

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3b
Sme Calculator3 marks

Hence solve, in the range 0 less or equal than space x less or equal than 2 straight pi, the equation

sec x cosec xnegative 5 equalscosec 2 x

giving your answers correct to 3 significant figures.

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4a
Sme Calculator3 marks

Show that the equation

tan2 x equals 6 sec x minus 10

can be rewritten in the form

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

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4b
Sme Calculator3 marks

Hence solve, in the range 0 less or equal than space x less or equal than 2 straight pi, the equation

tan2 x equals 6 sec x minus 10

giving your answers correct to 3 significant figures.

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5
Sme Calculator5 marks

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

(i)

state the range of possible values of x,

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

 ,

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6a
Sme Calculator4 marks

Prove 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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6b
Sme Calculator2 marks

Explain why this is not true for negative 1 less or equal than space x less than 0.

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7
Sme Calculator5 marks
(i)
Sketch, in the interval negative 2 straight pi less or equal than theta less or equal than 2 straight pi, the graph of y equals 3 plus 2 cosec theta , include asymptotes and label the coordinates of all maximum and minimum points.

(ii)
Hence, deduce the number of solutions to the equation 3 plus 2 cosec theta equals 1 half in the interval  negative 2 straight pi less or equal than theta less or equal than 2 straight pi.

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8a
Sme Calculator3 marks

The function  straight f is defined as straight f open parentheses x close parentheses equals arc cos x , negative 1 less or equal than space x less or equal than 1, and the function  straight g  is such that  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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8b
Sme Calculator3 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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8c
Sme Calculator2 marks

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

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1a
Sme Calculator2 marks

Rewrite tan theta cosec theta as a single trigonometric function.

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1b
Sme Calculator3 marks

Hence solve, in the range negative straight pi less than theta less or equal than straight pi, the equation

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

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2
Sme Calculator6 marks

Solve, in the range 0 less or equal than theta less or equal than 2 straight pi, the equation

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

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3
Sme Calculator6 marks

Using the double angle formula sin 2 A identical to 2 sin A cos A,  find the solutions to the equation

sec x cosec xnegative 75 equals 5 cosec 2 x

in the range negative straight pi less than space x italic less or equal than straight pi.   Give your answers correct to 3 significant figures.

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4a
Sme Calculator3 marks

Show that the equation

2 cot2 x equals 1 minus 5cosec x

can be rewritten 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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4b
Sme Calculator3 marks

Hence solve, in the range 0 less or equal than space x less or equal than 2 straight pi, the equation

2 cot2 x equals 1 minus 5cosec x

giving your answers correct to 3 significant figures.

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5
Sme Calculator5 marks

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

(i)

state the range of possible values of x,

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

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6
Sme Calculator7 marks

Prove that for negative 1 less or equal than space x less or equal than 0,arc cos x equals straight pi minus arcsin square root of 1 minus x to the power of italic 2 end root .

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7
Sme Calculator5 marks
(i)

Sketch, in the interval negative 2 straight pi less or equal than theta less or equal than 2 straight pi, the graph of  y equals negative 5 plus 1 halfsec theta  , include asymptotes and label the coordinates of all maximum and minimum points.

(ii)
Hence deduce the range of values for k for which the equation   negative 5 plus 1 halfsec theta equals straight k   has no solutions.

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8a
Sme Calculator3 marks

The function  straight f  is defined as straight f open parentheses x close parentheses equals arctan x ,  x element of straight real numbers  , and the function  straight g  is such that  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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8b
Sme Calculator3 marks

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

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8c
Sme Calculator2 marks

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

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1
Sme Calculator5 marks

Solve, in the range negative straight pi less than theta less or equal than straight pi, the equation

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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2
Sme Calculator6 marks

Solve, in the range 0 less or equal than theta less or equal than 2 straight pi, the equation

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

Leaving your answers as exact values.

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3
Sme Calculator6 marks

Using the double angle formulae sin 2 A identical to 2 sin A cos A and cos 2 A identical to cos squared A space minus space sin squared A,  find the solutions to the equation

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

in the range negative straight pi less than space x italic less or equal than straight pi.   Give your answers correct to 3 significant figures.

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4
Sme Calculator6 marks

Solve, in the range 0 less or equal than space x less or equal than 2 straight pi, the equation

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

Leaving your answers as exact values.

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5
Sme Calculator5 marks

Given that x satisfies the equation  ar c tan x equals k, where    negative straight pi over 2 less or equal than space k less or equal than 0

(i)

state the range of possible values of x,

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

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6
Sme Calculator7 marks

Prove that for x less or equal than negative 1,

arcsin 1 over x equals negative arccos open parentheses negative fraction numerator square root of x squared minus 1 end root over denominator x end fraction close parentheses

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7a
Sme Calculator3 marks

Sketch, in the interval negative 2 straight pi less or equal than theta less or equal than straight pi, the graph of  y equals 2 plus 3 space sec open parentheses theta plus straight pi over 2 close parentheses , include asymptotes and label the coordinates of all maximum and minimum points.

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7b
Sme Calculator4 marks

Deduce 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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8a
Sme Calculator4 marks

The function  straight f  is defined as  straight f space colon space x rightwards arrow from bar arcsin space x,  negative 1 less or equal than space x less or equal than 1 , and the function  straight g  is such that 

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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8b
Sme Calculator2 marks

Define the inverse function straight g to the power of negative 1 end exponent in the form  straight g to the power of negative 1 space end exponent colon space x italic rightwards arrow from bar… 

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8c
Sme Calculator3 marks

Over the same domain as  straight g, the function straight h is defined as  straight h space colon space x italic rightwards arrow from bar p italic space 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 two functions’ common domain, determine the values of p and q.

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