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A LevelMaths: Pure 3CIEExam Questions3. Trigonometry3.1 Reciprocal Trigonometric Functions

Reciprocal Trigonometric Functions (CIE A Level Maths: Pure 3)

Exam Questions

2 hours24 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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2
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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3
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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4
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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5
Sme Calculator3 marks

Show that

      sec2 theta sin theta identical totan theta sec theta

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6
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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7
Sme Calculator3 marks

Show that

         cot x cosec x sec x identical to 1 pluscot2 x

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8
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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9a
Sme Calculator2 marks

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

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9b
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

cosec 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
(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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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
(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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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 minus 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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5a
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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5b
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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