Edexcel International A Level Maths: Pure 3

Topic Questions

2.1 Reciprocal & Inverse Trigonometric Functions

1
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2 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
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2 marks

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

2b
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2 marks

Hence sketch the graph of y equalsarccos theta.

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3
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3 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
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4 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
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4 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
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3 marks

Show that

      sec2 theta sin theta identical totan theta sec theta

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7a
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2 marks

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

7b
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2 marks

Hence sketch the graph of y equals arc sin theta.

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8
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3 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
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3 marks

Show that

         cot x cosec x sec x identical to 1 pluscot2 x

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10
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4 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
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2 marks

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

11b
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2 marks

Hence sketch the graph of y equals arc tan theta.

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12a
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2 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.

12b
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2 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
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2 marks

Rewrite tan theta cosec theta as a single trigonometric function.

1b
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3 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
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6 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
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6 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
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3 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

4b
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3 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
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5 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
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7 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
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5 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
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3 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.

8b
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3 marks

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

8c
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2 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
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2 marks

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

sec theta cot theta identical tocosec theta.  

1b
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3 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
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2 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

2b
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4 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
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3 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.

3b
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3 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
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3 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

4b
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3 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
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5 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
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4 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.

6b
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2 marks

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

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7
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5 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
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3 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.

8b
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3 marks

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

8c
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2 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
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5 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
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6 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
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6 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
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6 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
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5 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
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7 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
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3 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.

7b
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4 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
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4 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.

8b
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2 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… 

8c
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3 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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