IBMathsDPAnalysis & Approaches (AA)HLExam Questions3. Geometry & TrigonometryInverse & Reciprocal Trigonometric Functions

Inverse & Reciprocal Trigonometric Functions (DP IB Analysis & Approaches (AA): HL): Exam Questions

3 hours29 questions
Medium
Hard
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1a
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1 mark

State the value ofarctan space open parentheses square root of 3 close parentheses. 

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1b
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6 marks

If arccos space x equals pi over 6 find 

(i) the exact value of arcsin space x.  

(ii) the exact value of sec left parenthesis arccos space x right parenthesis.

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2
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6 marks

Find the exact values of the following expressions:

(i) cosec space open parentheses straight pi over 3 close parentheses plus tan space open parentheses straight pi over 6 close parentheses 

(ii) 3 space sin space open parentheses straight pi over 4 close parentheses minus cot space open parentheses straight pi over 3 close parentheses

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3a
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2 marks

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

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3b
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5 marks

Given that cot space straight theta equals 9 over 7 and pi less or equal than theta less or equal than fraction numerator 3 straight pi over denominator 2 end fraction comma find the values of cos space theta comma space sin space theta and tan space theta.

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4
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5 marks

Solve tan squared space x equals sec space x plus 11 for 0 less or equal than x less or equal than pi.

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5a
3 marks

Show that the equation

sec space straight theta minus 5 space cos space straight theta equals 2 square root of 2

can be rewritten as 

5 space cos squared space theta plus 2 square root of 2 space cos space theta minus 1 equals 0

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5b
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3 marks

Hence, solve the equation sec space theta minus 5 space cos space theta equals 2 square root of 2 for all values of theta in the interval negative pi less or equal than theta less or equal than straight pi over 2.

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6a
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3 marks

A functionspace f can be defined byspace f left parenthesis x right parenthesis equals space 3 x minus 5 x space arcsin left parenthesis x right parenthesis, where negative 1 less or equal than x less or equal than 1. 

Sketch the graph ofspace f indicating clearly any intercepts with the coordinate axes and the coordinates of any local maximum or minimum points.

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6b
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2 marks

State the domain and range ofspace f.

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6c
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3 marks

Solve the inequality 3 x minus 5 x space arcsin space left parenthesis x right parenthesis greater than negative 2.

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7a
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3 marks

The functionspace f is defined asspace f left parenthesis x right parenthesis equals arccos space x comma negative 1 less or equal than x less or equal than 1 comma and the function g is such that g left parenthesis x right parenthesis equals f left parenthesis 3 x right parenthesis.

Sketch the graph of y equals f left parenthesis x right parenthesis and state the range ofspace f.

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7b
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3 marks

Sketch the graph of y equals g left parenthesis x right parenthesis and state the domain of g.

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7c
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2 marks

Find the inverse function g to the power of negative 1 end exponent left parenthesis x right parenthesis and state its domain.

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8a
2 marks

Show that  sec space theta space cot space theta identical to cosec space theta.

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8b
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3 marks

Hence solve in the range 0 less or equal than theta less or equal than 2 pi,  the equation sec space theta space cot space theta equals negative 2 

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9a
3 marks

Show that the equation 

tan to the power of 2 space end exponent x equals 6 space sec x minus 10

can be rewritten in the form 

left parenthesis sec space x minus 3 right parenthesis squared equals 0

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9b
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3 marks

Hence, solve the equation tan squared space x equals 6 space sec space x minus 10  in the range 0 less or equal than x less or equal than 2 pi.

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10a
3 marks

Show that the equation 

cot squared space x equals 9 minus 3 space cosec space x

can be rewritten in the form 

left parenthesis cosec space x minus 2 right parenthesis left parenthesis cosec space x plus 5 right parenthesis equals 0.

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10b
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3 marks

Hence, solve the equation cot squared space x equals 9 minus 3 space cosec space x in the interval negative 180 degree less or equal than x less or equal than 180 degree. Give your answers correct to 1 decimal place.

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1
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6 marks

If x equals sin open parentheses fraction numerator 2 pi over denominator 3 end fraction close parentheses find

(i) the exact value of cos open parentheses 3 arccos x close parentheses.

(ii) the exact value of cos open parentheses arc sin x close parentheses.

 

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2
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6 marks

Find the exact values for the following expressions

(i) space sec open parentheses pi over 4 close parentheses cos open parentheses pi over 6 close parentheses

(ii)      fraction numerator 2 tan squared open parentheses pi over 3 close parentheses over denominator cosec open parentheses pi over 6 close parentheses end fraction.

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3a
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2 marks

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

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3b
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4 marks

Given that space 2 cosec theta equals negative 9 over 4 and negative pi less than straight theta less than negative pi over 2,  find the exact value of cot space theta.

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4a
4 marks

Show that the equation 33 plus fraction numerator 6 space cos space x over denominator cot squared space x end fraction equals square root of 2 open parentheses 11 space sec space x plus 2 close parentheses   can be rewritten in the form a cos squared x plus b space cos space x plus c equals 0 .

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4b
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3 marks

Hence solve 33 plus fraction numerator 6 space cos space x over denominator cot squared space x end fraction equals square root of 2 open parentheses 11 sec space x plus 2 close parenthesesfor the interval negative fraction numerator 3 pi over denominator 2 end fraction less or equal than x less or equal than pi .

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5a
5 marks

Show that the equation

cot space 2 theta space sin space theta plus fraction numerator 8 over denominator sec space theta end fraction equals sin space theta

can be rewritten as  

tan squared theta plus 2 space tan space theta minus 17 equals 0.

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5b
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3 marks

Hence solve the equation cot space 2 theta space sin space theta plus fraction numerator 8 over denominator sec space theta end fraction equals sin space theta for all values of theta in the interval negative straight pi less or equal than theta italic less or equal than pi over italic 2.

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6a
3 marks

Consider the function f left parenthesis x right parenthesis equals 1 half arccos space x plus 1,  where negative 1 less or equal than x less or equal than 1.

Sketch the graph of f indicating clearly any intercepts with the coordinate axes and any maximum and minimum values.

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6b
1 mark

Write down the domain of f to the power of negative 1 end exponent left parenthesis x right parenthesis .

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6c
2 marks

Find an expression for f to the power of negative 1 end exponent open parentheses x close parentheses .

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7a
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2 marks

Consider the function  f left parenthesis x right parenthesis equals arccos space x comma space minus 1 less or equal than x less or equal than 1.

State whether the function is f even, odd or neither. Give a reason for your answer.

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7b
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2 marks

A second function g is such that g left parenthesis x right parenthesis equals 3 f left parenthesis x right parenthesis plus a , where g left parenthesis x right parenthesis is an odd function.

Find the value of a.

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7c
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3 marks

Sketch the graph of g left parenthesis x right parenthesis  and state the range of g. 

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7d
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2 marks

State whether g to the power of negative 1 end exponent open parentheses x close parentheses will also be odd. Give a reason for your answer.

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8a
2 marks

Show that fraction numerator cot space theta space sec squared theta over denominator cosec space theta end fraction identical to sec theta .

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8b
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3 marks

Hence solve in the range negative pi less or equal than theta less or equal than 2 pi ,  the equation  fraction numerator square root of 3 cot italic space theta italic space sec to the power of italic 2 italic space theta over denominator cosec space theta end fraction equals 2  .

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9a
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2 marks

Consider the function defined by f left parenthesis x right parenthesis equals sec open parentheses 2 arcsin straight x close parentheses. 

Find the domain of f.

 

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9b
6 marks

Show that f left parenthesis x right parenthesis  can be written as fraction numerator 1 over denominator 1 minus 2 x squared end fraction  for all x in its domain.

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10a
4 marks

Show that the equation

3 open parentheses fraction numerator 1 over denominator cot squared x end fraction minus fraction numerator 3 over denominator cos space x end fraction close parentheses equals 8 space sec space x plus 25

can be written in the form

open parentheses a space sec space x plus b close parentheses open parentheses sec space x plus c close parentheses equals 0.

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10b
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3 marks

Hence solve the equation 3 open parentheses fraction numerator 1 over denominator cot squared x end fraction plus fraction numerator 3 over denominator cos space x end fraction close parentheses equals 8 space sec space x plus 25 in the interval negative 270 degree less or equal than x less or equal than 90 degree. Give your answers correct to 1 decimal place.

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1a
4 marks

Show that fraction numerator cot space x over denominator sec space x open parentheses 1 minus sin space x close parentheses end fraction identical to 1 plus fraction numerator 1 over denominator sin space x end fraction comma space x not equal to fraction numerator n straight pi over denominator 2 end fraction comma space n element of straight integer numbers.

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1b
4 marks

Hence determine the set of values for k for which fraction numerator cot space x over denominator sec space x open parentheses 1 minus sin italic space x close parentheses end fraction equals k has no real solutions.

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2a
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4 marks

Find the exact value for the expression cot squared open parentheses pi over 6 close parentheses plus cosec open parentheses fraction numerator 2 pi over denominator 3 end fraction close parentheses.

Give your answer in the form a plus fraction numerator b square root of c over denominator d end fraction where a comma b comma c comma d element of straight integer numbers.

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2b
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3 marks

Find the exact value for the expression fraction numerator sec open parentheses fraction numerator 2 pi over denominator 3 end fraction close parentheses over denominator cot open parentheses pi over 4 close parentheses end fraction .

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3a
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2 marks

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

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3b
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5 marks

Given that sec space 2 theta equals 8 over 7and  negative 2 pi less or equal than theta less or equal than negative fraction numerator 3 pi over denominator 2 end fraction,  find the value of sin space theta .

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4a
4 marks

Show that the equation

 5 sin space x minus 9 space cos space x equals fraction numerator 2 space cosec space x over denominator 1 plus tan squared x end fraction 

can be rewritten as

open parentheses a space sin space x plus cos space x close parentheses open parentheses sin space x minus b space cos space x close parentheses equals 0

 

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4b
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3 marks

Hence solve the equation 5 space sin space x minus 9 space cos space x equals fraction numerator 2 space cos e c space x over denominator 1 plus tan squared x end fraction comma space minus pi less or equal than x less or equal than pi.

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5a
4 marks

Consider the function f left parenthesis x right parenthesis equals cos squared open parentheses 2 space arcsin open parentheses x close parentheses close parentheses,0 less or equal than x less or equal than a  where a is a constant.

Given that f has an inverse, find an expression for f to the power of negative 1 end exponent open parentheses x close parentheses.

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5b
4 marks

Show that cos squared open parentheses 2 space arcsin open parentheses x close parentheses close parentheses equals 4 x squared minus 4 x squared plus 1 for negative 1 less or equal than x less or equal than 1.

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5c
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2 marks

Given that f has an inverse, find the maximum value for a .

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6a
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3 marks

A function f can be defined by f open parentheses x close parentheses equals 1 half tan open parentheses arcsin open parentheses x close parentheses close parentheses plus 5.

Sketch the graph of f indicating clearly any intercepts with the coordinate axes and asymptotes.

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6b
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5 marks

Solve the inequality 1 half tan open parentheses arcsin open parentheses x close parentheses close parentheses plus 5 greater than 2. 

Give your answer in the form  fraction numerator a square root of b over denominator c end fraction less than x less than d where a comma b comma c comma d element of straight integer numbers.

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7a
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2 marks

Consider the equation fraction numerator 1 minus cos squared theta over denominator cot space theta end fraction equals negative fraction numerator 1 over denominator cosec squared space theta end fraction.

Explain why the equation is undefined when theta equals 1 half k pi for any k element of straight integer numbers.

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7b
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6 marks

Solve the equation fraction numerator 1 minus cos squared theta over denominator cot theta end fraction equals negative fraction numerator 1 over denominator cos e c squared theta end fraction comma space for space minus pi less or equal than theta less or equal than 2 pi.

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8
8 marks

Show that sin squared open parentheses 1 half arctan open parentheses x close parentheses close parentheses equals fraction numerator 1 plus x squared minus square root of 1 plus x squared end root over denominator 2 open parentheses 1 plus x squared close parentheses end fraction comma space space x element of straight real numbers.

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9
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6 marks

Solve the equation 2 sin space x plus fraction numerator cot space x over denominator sec space x end fraction equals 3 space cos space xfor the interval negative 90 degree less than x less than 90 degree.

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