DP IB Maths: AA SL

Topic Questions

3.4 Further Trigonometry

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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6a
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6 marks

Wynken, Blynken and Nod are three mathematics students.  While the three are revising trigonometry, Nod sets the following problem for his two companions:

“ABC is a triangle with AC equals 8.1 space cm comma space space BC equals 9.8 space cm and straight A straight B with hat on top straight C equals 47 degree. To three significant figures, what is the size of the largest angle in the triangle?”

Wynken and Blynken set to work, and several minutes pass. “70.8 degrees,” states Wynken confidently.  “118 degrees,” insists Blynken a moment later.

Demonstrate that Wynken’s and Blynken’s responses may both be correct answers to the problem Nod has set them.

6b
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1 mark

Suggest an additional piece of information that Nod could provide, that would allow his problem to have a single unique solution.

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7
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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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8a
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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.

8b
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3 marks

Find the perimeter of sector OAB.

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9
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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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10a
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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

10b
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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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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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6
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4 marks

Two non-congruent triangles both have sides AB space equals space 5.3 space cm, BC space equals space 6.4 space cm and straight A straight C with hat on top straight B space equals space 38 degree.

Show that the angle BA ̂ C for one of the triangles is 132°, to 3 significant figures.

Find the angle AB ̂ C for the other triangle.

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7
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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 is the smallest angle in the triangle.

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8a
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5 marks

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

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

(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.
8b
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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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9
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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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10
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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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11a
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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.

11b
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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.

11c
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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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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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5a
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5 marks

Triangle ABC is such that the length of side AB is x units, the length of side BC is y units, and straight B straight A with hat on top straight C space equals space theta is an acute angle.

Use a diagram to show that if (and only if) x space sin space theta less than y less than x then there are two triangles, ABC subscript 1 and ABC subscript 2, which satisfy the conditions above, where straight C subscript 1 and straight C subscript 2 are points such that AC subscript 1 greater than AC subscript 2.

5b
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5 marks

Given that x space sin space theta less than y less than x, let angle straight A stack straight C subscript 1 with hat on top straight B be denoted by ∅.

(i)
Write down an expression for ∅ in terms of x comma space y and θ.

(ii)
Write down expressions for angles straight A straight B with hat on top straight C subscript 1 comma space straight A stack straight C subscript 2 with hat on top straight B and straight A straight B with hat on top straight C subscript 2 in terms of θ and ∅.
5c
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5 marks

Show that the difference between the areas of triangles ABC subscript 1 and ABC subscript 2 is equal to

1 half y squared space sin space 2 ϕ

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6
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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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7
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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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8
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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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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 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

9b
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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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