Further Trigonometry (DP IB Maths: AI HL)

Exam Questions

4 hours35 questions
1
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5 marks

Complete the table.

Degrees Radians sin cos tan
  straight pi over 6   fraction numerator square root of 3 over denominator 2 end fraction  
45 degree     fraction numerator 1 over denominator square root of 2 end fraction  
60 degree straight pi over 3      
  fraction numerator 2 straight pi over denominator 3 end fraction fraction numerator square root of 3 over denominator 2 end fraction    
270 degree        

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2
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3 marks

Given that  sin space theta equals 3 over 5 space,  where  pi over 2 less than theta less than pi,  find the possible values of cos theta and tan theta.

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3
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7 marks

The following triangle shows triangle ABC, with  AB space equals space 3 a comma space space BC space equals space a space spaceand  AC space equals 7.

q3-3-4-medium-ib-aa-sl-maths

Given that cos space straight A straight B with hat on top straight C equals 1 half,  find the area of the triangle.  Give your answer in the form fraction numerator p square root of 3 over denominator r end fraction,   where  p comma space q element of straight real numbers..

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4a
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3 marks

The following triangle shows triangle ABC, with  AB space equals 15 comma space space AC space equals 20 comma space space BC space equals x.

q4-3-4-medium-ib-aa-sl-maths

Given that cos space straight B straight A with hat on top straight C equals 2 over 3,  find the value of  sin space straight B straight A with hat on top straight C.

4b
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3 marks

Find the exact area of triangle ABC.

4c
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3 marks

By finding the value of x, show that triangle ABC is isosceles.

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5
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4 marks

A sector of a circle, OPQ, is such that it has radius 3.4 cm and the angle at its centre, O, is fraction numerator 3 straight pi over denominator 4 end fraction  radians.

(i)
Find the length of the arc PQ.

(ii)
Find the area of the sector OPQ.

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6a
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5 marks

The diagram below shows the sector of a circle O A B.

q6-medium-3-4-ib-ai-hl-furthertrigonometry

i)
Find the area of the sector O A B, giving your answer to 3 significant figures.

ii)
Find the area of the triangle O A B, giving your answer to 3 significant figures.

iii)
Find the area of the shaded segment, giving your answer to 3 significant figures.
6b
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3 marks
i)
Find the length of the arc A B.

ii)
Find the perimeter of the sector O A B.

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7
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3 marks

The canopy of a parachute and the outermost connecting cords form a sector of a circle as shown in the diagram below, with the parachutist modelled as a particle at point O.

q9a-3-4-medium-ib-aa-sl-maths

The area of the sector OAB is  fraction numerator 81 straight pi over denominator 200 end fraction straight m squared.

Find the length of one of the connecting cords on the parachute.

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8
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3 marks

A plastic puzzle piece is in the form of a prism with a cross-section that is the sector of a circle, as shown in the diagram below.  The radius of the sector is 8 space cm, and the angle at the centre is 1.2 space radians.

The height of the puzzle piece is 2 space cm.

q10-3-4-medium-ib-aa-sl-maths

(i)
Work out the area of the cross-section.

(ii)
Hence, or otherwise, work out the volume of the puzzle piece.

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9a
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2 marks

The circle sector OAB is shown in the diagram below.

The angle at the centre is  straight pi over 3 radians, and the line segments OC and BC have lengths of 8 space cm and p space cm respectively.

Additionally, C D is parallel to AB, so that A D equals B C and O D equals O C.

q11a-3-4-medium-ib-aa-sl-maths

Show that the area of the sector O A B is straight pi over 6 left parenthesis p plus 8 right parenthesis squared space cm squared.

9b
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2 marks

Show that the area of the triangle O C D is 16 square root of 3 space cm squared.

9c
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4 marks

Given that the area of the shaded shape A B C D is open parentheses fraction numerator 50 straight pi over denominator 3 end fraction minus 16 square root of 3 close parentheses space cm squared comma find the value of p.

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10
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5 marks

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

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11
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6 marks

A right-angled triangle has hypotenuse 8 cm. One of its other sides is 5 cm.

Find exact values for sin space theta, cos space theta and tan space theta, where theta is the smallest angle in the triangle.

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12a
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1 mark

The graph below shows the curve with equation y space equals space sin space 2 x in the interval negative 60 degree space less or equal than space x space less or equal than 270 degree.

q12a-medium-3-4-ib-ai-hl-furthertrigonometry

Point A has coordinates left parenthesis long dash 45 degree comma space long dash 1 right parenthesis and is the minimum point closest to the origin.
Point B is the maximum point closest to the origin. State the coordinates of B.

12b
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2 marks

A straight line with equation space y space equals negative space 1 halfmeets the graph of y space equals space sin space 2 x at the three points P comma space Q and R, as shown in the diagram.
Given that point P has coordinates left parenthesis long dash 15 degree comma space minus 1 half right parenthesis, use graph symmetries to determine the coordinates of Q and R.

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13
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4 marks
i)
Sketch the graph of y space equals space cos space left parenthesis space theta space plus space 30 degree right parenthesis in the interval long dash 180 degree space less or equal than space theta space less or equal than 360 degree.

ii)
Write down all the values where cos left parenthesis theta space plus space 30 degree right parenthesis equals 0 in the given interval.

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1
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5 marks

Complete the following table. In all cases the values for the angle should be given between 0 and 360° or 0 and 2 straight pi radians, as appropriate.

Degrees Radians sin cos tan
45 degree       1
    fraction numerator square root of 3 over denominator 2 end fraction 1 half  
150 degree   1 half    
    negative 1    
  fraction numerator 7 straight pi over denominator 4 end fraction      

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2
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4 marks

Given that sin space theta equals negative 12 over 13 ,  find the possible values of cos space theta and the corresponding values of tan space theta.

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3
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6 marks

The following diagram shows triangle ABC, with BC equals 8 and  AC equals x.

q3-3-4-hard-ib-aa-sl-maths

Given that tan space straight A straight C with hat on top straight B equals space 5 over 12   and that the area of triangle ABC is equal to 20 units squared, find the value of x.

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4a
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6 marks

The following diagram shows triangle ABC, with AB equals 4 comma space space BC equals 5 and  AC equals x.

q4a-3-4-hard-ib-aa-sl-maths

Given that  cos space straight A straight B with hat on top straight C equals 3 over 4,  find the exact area of triangle ABC.

4b
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4 marks

Find the exact perimeter of triangle ABC.

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5
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6 marks

A sector of a circle, OPQ, is such that the angle at its centre, O, is fraction numerator 5 straight pi over denominator 6 end fraction radians.

The area of sector OPQ in cm squared is one-fifth of the length of the arc PQ in cm.

(i)
Show that the radius of the sector is equal to 0.4 cm, and hence

(ii)
find the area of sector OPQ and the length of arc PQ.

Give your answers in part (ii) correct to 3 significant figures.

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6
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7 marks

The lengths of two sides in a right-angled triangle are 9 cm and 12 cm.

Find the possible values of sin space theta, and the corresponding values of cos space theta and tan space theta, where theta is the smallest angle in the triangle.  All your answers should be given as exact values.

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7a
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5 marks

The diagram below shows the sector of a circle OAB, with centre O and radius 9.8 cm. The angle at the centre of the sector, straight A straight O with hat on top straight B, is 0.85 radians.

q8a-3-4-hard-ib-aa-sl-maths

Find the area of the shaded segment, bounded by arc AB and chord AB. Give your answer correct to 3 significant figures.

7b
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3 marks

Find the perimeter of sector OAB.

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8
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5 marks

A games design company produces a popular game called ‘Inconsequential Endeavour’. Each game set includes solid plastic game pieces which are in the form of a right prism with a cross-section that is the sector of a circle, as shown in the diagram below. The angle at the centre of the sector is 1.05 space radians, and the height of the game piece is 4 space mm.

q9-3-4-hard-ib-aa-sl-maths

Given that the volume of the game piece is 0.412 space cm cubed, work out the radius of the sector. Give your answer correct to 3 significant figures.

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9a
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6 marks

The diagram below shows the sector of a circle OAB with centre O. The angle at the centre of the sector, straight A straight O with hat on top straight B, is fraction numerator 5 straight pi over denominator 6 end fraction radians. Point M is the midpoint of line segment OA, and the shaded region is the combination of triangle ABM with the region enclosed by the arc AB and the chord AB.

q10a-3-4-hard-ib-aa-sl-maths

Show that the ratio of the area of triangle OMB to the area of the shaded region may be expressed as

1 space colon space open parentheses fraction numerator 10 pi over denominator 3 end fraction space minus 1 close parentheses

9b
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3 marks

Given that the area of the shaded region is equal to 30 pi minus 9 space units squared, find the exact area of triangle OAB.

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10a
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3 marks

A graph has the equation y space equals space cos space 3 x for the interval negative pi over 3 less or equal than x less or equal than 2 over 3 pi.

Sketch the graph on the axes below.

q10-hard-3-4-ib-ai-hl-furthertrigonometry

10b
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5 marks

A straight line with equation y space equals 1 half intersects the graph of y space equals space cos space 3 x.

i)
Sketch the line y space equals space 1 half on to the same set of axes.
ii)
Find the coordinates of the points of intersection between y space equals space cos space 3 x space and space y space equals space 1 half.

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11
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5 marks
i)
Sketch the graph of y space equals space sin space left parenthesis 2 theta space long dash space 60 right parenthesis in the interval long dash 180 degree less or equal than space theta space less or equal than space 180 degree.

ii)
Write down all the values for theta, where  space y space equals space sin left parenthesis 2 theta space long dash space 60 right parenthesis degree space equals space 0 spacein the given interval.

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12a
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2 marks

Use the fact that

16 x cubed space minus space 12 x squared space minus 4 x plus 3 space equals space open parentheses 4 x minus 3 close parentheses open parentheses 4 x squared minus 1 close parentheses

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

12b
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7 marks

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

16 space sin cubed space 3 theta space long dash space 12 space sin squared space 3 theta space long dash space 4 sin space 3 theta space plus 3 space equals space 0

in the interval 0 space 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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1a
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3 marks

In each of the following, theta is an angle measured in radians such that 0 less than theta less than straight pi over 2.

Given that sin space theta equals p,  write down expressions for sin space left parenthesis pi minus theta right parenthesis comma space cos space left parenthesis pi minus theta right parenthesis and tan space left parenthesis pi minus theta right parenthesis.

1b
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3 marks

Given that cos space theta equals q,  write down expressions for sin space left parenthesis pi plus theta right parenthesis comma space cos space left parenthesis pi plus theta right parenthesis and tan space left parenthesis pi plus theta right parenthesis.

1c
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4 marks

Given that tan space theta equals r, write down expressions for sin space left parenthesis 2 pi minus theta right parenthesis comma space cos space left parenthesis 2 pi minus theta right parenthesis and tan space left parenthesis 2 pi minus theta right parenthesis.

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2
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5 marks

Given that tan space theta equals negative 4 over 3 find the possible values of sin space theta and the corresponding values of cos space theta.

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3
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8 marks

The following diagram shows triangle ABC, with  AC equals square root of 41.

q3-3-4-very-hard-ib-aa-sl-maths

Given that tan space straight A straight B with hat on top straight C equals negative 12 over 5 and that the ratio of the length of side AB to the length of side BC is 10 space colon 13, find the exact area of triangle ABC.

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4
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6 marks

A sector of a circle, OPQ, is such that the angle at its centre, O, is fraction numerator 11 straight pi over denominator 14 end fraction radians.

Given that the area of sector OPQ in cm squared is equal to the length of the arc PQ in mm, find

(i)
the area of sector OPQ , and

(ii)
the length of arc PQ

giving your answers correct to 3 significant figures.

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5
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9 marks

The lengths of two sides in a right-angled triangle are x and y, with square root of 2 x less than y.

Find the possible values of sin space theta, and the corresponding values of cos space theta and tan space theta, where theta is the smallest angle in the triangle.  All your answers should be given in terms of x and y.

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6
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8 marks

The diagram below shows the sector of a circle OAB, with centre O. The angle at the centre of the sector, straight A straight O with hat on top straight B, is 1.3 radians.  The shaded region in the diagram is the segment bounded by the arc AB and the chord AB.

q7-3-4-very-hard-ib-aa-sl-maths

Given that the difference between the perimeter of sector OAB and the perimeter of  triangle OAB is 1.05 cm, find the area of the shaded region. Give your answer correct to 3 significant figures.

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7
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8 marks

A games design company produces a popular game called ‘Nugatory Enterprise’. Each game set includes plastic game pieces which are in the form of a right prism with a cross-section that is the sector of a circle, as shown in the diagram below.  The angle at the centre of the sector is 0.9 space radians, and the height of the game piece is 6 mm.

q8-3-4-very-hard-ib-aa-sl-maths

The game pieces are hollow, with a top (which is a cross-section of the prism) and three sides, but no bottom.

Given that the external surface area of the game piece is 4.26 space cm squared, work out the interior volume of the game piece giving your answer correct to 3 significant figures. You may ignore the thickness of the top and sides in your calculations.

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8a
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6 marks

The diagram below shows the sector of a circle OAB with centre O. The angle at the centre of the sector, straight A straight O with hat on top straight B, is straight pi over 6 radians.  Point P is a point on line segment [OB] such that OP equals k cross times OB, where k is a constant with 0 less than k less than 1. The shaded region in the diagram is the combination of triangle ABP with the region enclosed by the arc AB and the chord AB.

q9-3-4-very-hard-ib-aa-sl-maths

If the area of triangle OAP is denoted by a and the area of the shaded region is denoted by b, show that

k space equals space open parentheses fraction numerator a over denominator a space plus space b end fraction close parentheses straight pi over 3

8b
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6 marks

Given that  OP equals 7 space cm and that the area of triangle OAP is one half the area of sector OAB, show that the exact length of chord AB in centimetres is given by

AB space equals space 42 over straight pi square root of 2 minus square root of 3 end root

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9
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4 marks
i)
Sketch the graph of y space equals space 4 space sin left parenthesis theta space plus space 45 right parenthesis space long dash space fraction numerator 4 over denominator square root of 2 end fraction in the interval negative 180 degree space less or equal than theta less or equal than 360 degree.
ii)
Write down all the values for theta, where 4 space sin left parenthesis theta space plus space 45 right parenthesis space minus space begin inline style fraction numerator 4 over denominator square root of 2 end fraction end style equals 0 in the given interval.

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10
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4 marks

Solve the equation

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


in the interval negative 180 degree space less or equal than x less or equal than space 90 degree.

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