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IBMaths: AA HLDPExam Questions3. Geometry & Trigonometry3.6 Trigonometric Equations & Identities

Trigonometric Equations & Identities (DP IB Maths: AA HL)

Exam Questions

3 hours25 questions
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1
Sme Calculator6 marks

The value of sin space alpha equals 3 over 7  for  0 less or equal than alpha less or equal than pi over 2.  Find:

(i)
cos space alpha
(ii)
sin space 2 alpha
(iii)
cos space 2 alpha
(iv)
tan space 2 alpha.

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2
Sme Calculator6 marks

The value of cos space B equals space 1 fifth,  for  fraction numerator 3 straight pi over denominator 2 end fraction less or equal than B less or equal than 2 pi.  Find:

(i)
cos space 2 B
(ii)
sin space 2 B
(iii)
tan space 2 B.

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3
Sme Calculator4 marks

An angle M has the properties such that sin space M space equals space r and sin space 2 M space equals space s.  Find, in terms of r and s comma an expression for:

(i)
cos space M
(ii)
tan space M.

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4
Sme Calculator3 marks

Solve the equation 2 space sin space 2 theta equals 1 for  0 degree less or equal than theta less or equal than 360 degree.

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5
Sme Calculator5 marks

Solve the equation 2 space sin space x equals fraction numerator 1 over denominator sin space x end fraction for  0 degree less or equal than x less or equal than 360 degree.

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6a
Sme Calculator2 marks

Show that left parenthesis x plus 1 right parenthesis left parenthesis x minus 2 right parenthesis left parenthesis x minus 3 right parenthesis equals x cubed minus 4 x squared plus x plus 6.

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6b
Sme Calculator5 marks

Use your result from part (a) to solve the equation

tan cubed space x minus 4 tan squared space x plus tan space x plus 6 equals 0

in the interval 0 degree less or equal than x less or equal than 360 degree.

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7a
Sme Calculator2 marks

Show that the equation 2 space sin squared space x plus 3 space cos space x equals 0 can be written in the form a space cos squared space x plus b space cos space x plus c equals 0, where a comma space b and c are integers to be found.

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7b
Sme Calculator3 marks

Hence, or otherwise, solve the equation 2 space sin squared space x plus 3 space cos space x space equals space 0
for negative 180 degree less or equal than x less or equal than 180 degree.

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8a
Sme Calculator1 mark

Show that the equation

2 space cos squared space x minus sin space x equals 1

can be written in the form

2 space sin squared space x plus sin space x minus 1 equals 0

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8b
Sme Calculator5 marks

Hence, solve the equation 2 space cos squared space x minus sin space x equals 1, for  0 less or equal than x less or equal than 4 pi

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9a
Sme Calculator4 marks

The graph below shows the function y equals f left parenthesis x right parenthesis where  f left parenthesis x right parenthesis equals cos space x space space for space minus pi less or equal than x less or equal than pi.

q3a-3-6-medium-ib-aa-sl-maths

The function g left parenthesis x right parenthesis is formed by translating the function f left parenthesis x right parenthesis 1 unit vertically downwards.

The function h left parenthesis x right parenthesis is formed by stretching the function f left parenthesis x right parenthesis by a factor of 1 half in the x direction. The domain of h left parenthesis x right parenthesis remains the same as f left parenthesis x right parenthesis.

(i)
Sketch the functions y equals h left parenthesis x right parenthesis and y equals g left parenthesis x right parenthesis.
(ii)
State the number of roots for g left parenthesis x right parenthesis.

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9b
Sme Calculator4 marks

Find the solutions to the equation cos space 2 x equals cos space x minus 1, for space minus pi less or equal than x less or equal than pi comma  and label them clearly on the graph of y equals f left parenthesis x right parenthesis given above.

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1a
Sme Calculator7 marks

The value of tan space alpha equals negative 3 over 2 for straight pi over 2 less or equal than alpha less or equal than pi.

Find

(i)
sin space alpha
(ii)
cos space alpha
(iii)
sin space 2 alpha
(iv)
cos space 2 alpha
(v)
tan space 2 alpha.

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1b
Sme Calculator2 marks

Use your results from part (a) to explain why pi less than 2 alpha less than fraction numerator 3 pi over denominator 2 end fraction must be true.

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2a
Sme Calculator3 marks

The value of cos space B equals x, for  pi less or equal than B less or equal than fraction numerator 3 pi over denominator 2 end fraction. 

Explain why

(i)
x less or equal than 0
(ii)
sin space B equals negative square root of 1 minus x squared end root.

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2b
Sme Calculator5 marks

Hence find the following in terms of x:

(i)
cos space 2 B
(ii)
sin space 2 B
(iii)
tan space 2 B.

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3
Sme Calculator6 marks

An angle M is such that sin space M equals p and cos space M equals q.  Show that

(i)
sin space 4 M equals 4 p q cubed minus 4 p cubed q
(ii)
cos space 4 M equals 8 q to the power of 4 minus 8 q squared plus 1.

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4
Sme Calculator5 marks

Solve the equation square root of 3 space cos space 2 theta equals 2 space cos squared space 2 theta in the interval  0 less or equal than theta less or equal than 360 degree.

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5
Sme Calculator5 marks

Solve the equation square root of 3 space tan space x equals fraction numerator 1 over denominator square root of 3 tan space x end fraction for 0 less or equal than x less or equal than 540 degree.

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6a
Sme Calculator2 marks

Use the fact that

 16 x cubed minus 12 x squared minus 4 x plus 3 equals left parenthesis 4 x minus 3 right parenthesis left parenthesis 4 x squared minus 1 right parenthesis

to fully factorise  16 x cubed minus 12 x squared minus 4 x plus 3.

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6b
Sme Calculator7 marks

Use your result from part (a) to solve the equation

16 space sin to the power of 3 space end exponent 3 theta minus 12 space sin to the power of 2 space end exponent 3 theta minus 4 space sin space 3 theta plus 3 equals 0

in the interval  0 less or equal than theta less or equal than pi over 2.  You should give your answers as exact values where possible.

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7
Sme Calculator6 marks

Solve the equation

sin squared space x equals space fraction numerator cos space x plus 4 over denominator 6 end fraction

in the interval negative pi less or equal than x less or equal than pi. Give your answers as exact values where possible.

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8a
Sme Calculator2 marks

Two functions, f and g, are defined by f left parenthesis x right parenthesis equals sin space x and  g left parenthesis x right parenthesis equals cos space 2 x.

Describe the single transformation of the graph of y equals cos space x that will produce the graph of  y equals g left parenthesis x right parenthesis.

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8b
Sme Calculator4 marks

On the same set of axes, sketch the graphs of y equals f left parenthesis x right parenthesis and y equals g left parenthesis x right parenthesis in the interval negative pi less or equal than x less or equal than 3 pi.

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8c
Sme Calculator6 marks

By using an appropriate trigonometric identity to solve the equation sin space x equals cos space 2 x in the interval  negative pi less or equal than x less or equal than 3 pi, determine the points of intersection of the two curves from your graph in part (b).  Label those points on your graph.

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1a
Sme Calculator3 marks

The value of tan space 2 alpha equals 2 over 5 for pi over 2 less or equal than 2 alpha less or equal than fraction numerator 3 pi over denominator 2 end fraction.

Find

(i)
sin space alpha
(ii)
cos space alpha

 

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1b
Sme Calculator6 marks

Hence show that

sin space alpha equals square root of fraction numerator k plus 5 square root of k over denominator 2 k end fraction end root space space space space space and space space space space cos space alpha equals negative square root of fraction numerator k minus 5 square root of k over denominator 2 k end fraction end root

where k is a positive integer to be determined, and use those results to find the exact value of tan space alpha.

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2
Sme Calculator7 marks

The value of sin space B equals x,  for  pi over 2 less than B less than pi.

Find the following in terms of x:

(i)
sin space 2 B
(ii)
cos space 2 B
(iii)
tan space 2 B.

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3a
Sme Calculator5 marks

It is given that cos space 6 x equals p.

Show that

tan space 3 x equals plus-or-minus square root of fraction numerator 1 minus p over denominator 1 plus p end fraction end root

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3b
Sme Calculator3 marks

For  0 less than x less than pi over 3,  determine the range of x values for which

(i)
the ‘plus’ version of the part (a) result should be used
(ii)
the ‘minus’ version of the part (a) result should be used
(iii)
the value of tan space 3 x is not defined.

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4
Sme Calculator7 marks

Solve the equation

sin space 2 theta minus sin space theta plus square root of 3 space cos space theta equals fraction numerator square root of 3 over denominator 2 end fraction

in the interval 0 less or equal than theta less or equal than 360 degree.

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5
Sme Calculator5 marks

Solve the equation

sin space 2 x space tan space 2 x equals fraction numerator 1 over denominator 2 space cos space 2 x end fraction

in the interval negative 90 degree less or equal than x less or equal than 90 degree.

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6
Sme Calculator6 marks

Solve the equation 

fraction numerator 5 space sin squared space x minus 2 over denominator cos space x end fraction equals 3 minus cos space x

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

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7a
Sme Calculator2 marks

Use the fact that

3 p cubed minus left parenthesis 7 minus 3 square root of 3 right parenthesis p squared minus left parenthesis 20 plus 7 square root of 3 right parenthesis p minus 20 square root of 3 equals left parenthesis 3 p plus 5 right parenthesis left parenthesis p squared plus left parenthesis square root of 3 minus 4 right parenthesis p minus 4 square root of 3 right parenthesis

to fully factorise  3 p cubed minus left parenthesis 7 minus 3 square root of 3 right parenthesis p squared minus left parenthesis 20 plus 7 square root of 3 right parenthesis p minus 20 square root of 3. end root

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7b
Sme Calculator8 marks

Two functions, f  and g, are defined by

f left parenthesis x right parenthesis equals 3 space tan cubed space 3 x minus left parenthesis 20 plus 7 square root of 3 right parenthesis space tan space 3 x space and space g left parenthesis x right parenthesis equals left parenthesis 7 minus 3 square root of 3 right parenthesis tan squared space 3 x plus 20 square root of 3 

for  negative pi over 6 less or equal than x less or equal than pi over 3.

Use an algebraic method along with your result from part (a) to determine the x-coordinates of the points of intersection of the curves y equals f left parenthesis x right parenthesis and  y equals g left parenthesis x right parenthesis.

Your solution should show clear algebraic working, and your answers should be given as exact values where possible.

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8a
Sme Calculator9 marks

Let OAB be an isosceles triangle with  OA equals OB and  straight A straight O with hat on top straight B equals straight theta.

If the length of line segment AB is denoted by p, and the area of triangle OAB is denoted by q, show that

cos space theta equals fraction numerator 1 minus m over denominator 1 plus m end fraction

where

m equals fraction numerator p to the power of 4 over denominator 16 q squared end fraction

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8b
Sme Calculator5 marks

The diagram below shows circle sector OAB with centre O and angle at the centre  straight A straight O with hat on top straight B equals theta.

q8a-3-6-very-hard-ib-aa-sl-maths

Given that the length of chord AB is 2 square root of 3 units, and that the area of triangle OAB is square root of 3 space units squared, find the area of sector OAB and the length of arc AB.

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