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## Trigonometry

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### What you will be practising:

Concept Examples

Pythagoras’ Theorem given a diagram. Given two sides in a right angled triangle, find

the third side. Includes when you need to find the hypotenuse and when you know the hypotenuse and need to find one of the other sides.

Pythagoras’ Theorem given no diagram and you need to draw one first.

1. A ship left a port and travelled 152 Km due

east and then 193 Km due north. How far was it from its starting point?

2. Two points A and B, are drawn on a

co-ordinate grid. A is (a, b) and B is (c, d). What is the distance between A and B? Pythagoras’ Theorem applied to 3-D

situations.

Find AG Trigonometry – finding the values of sines,

cosines and tangents and combinations of these functions.

Find: a) sin 141o, b) 9 tan 37o,

c) (sin 37.5o)2 + (cos 37.5o)2

Trigonometry – finding an angle given the value of sine, cosine or tangent of the angle.

sin x = 0.7382, find the value of x. Trigonometry – solving triangles.

Find x

z

3 m 4 m

A

B

C D

E

F

H

G 8.5 cm

7.2 cm 7.6 cm

x 8.7 cm

(4)

Trigonometry – more advanced problems. θ is one angle in a right angled triangle.

Cos θ = 12/13 and the hypotenuse is 65 cm

long. What is the length of the other two sides? Trigonometry - given no diagram and you need

to draw one first.

Michael is flying a kite on a string 26 metres

long. The angle of elevation of the kite is 42o.

Michael’s hand is 1.4 metres above the ground.

How high is the kite above the ground? Bearings – Solving problems using

trigonometry applied to bearings.

A ship sails from a port P on a bearing of 206o

for a distance of 56 Km to an island Q and

then changes direction to a bearing of 143o for

a distance of 98 Km to another port R. How far is the ship now east of its starting point and how far is it south of its starting point?

Trigonometry and Pythagoras’ Theorem to more complex problems.

A point P and two flagpoles are in a straight line.

P is positioned such that P and the tops of the flagpoles are also in a straight line. The flagpoles are 25 metres apart, the smaller is 18 metres high and the angle of elevation from

P to the tops of the flagpoles is 27.5o.

Find the height of the taller flagpole. Problems involving mixed trigonometry and

Pythagoras’ Theorem in 3-D The base ABCD of a right square based pyramid is laid horizontally on a table. The

vertex P is directly over the centre of the base. AB = 12 cm. AP = 22 cm.

Find:

a) The height of the pyramid.

b) The angle PDB.

c) The area of one of the sloping faces.

Finding sines, cosines and tangents and their

inverses over the range of 0o to 360o.

Give the angles, x, between 0o and 360o for

each of the following:

a) sin x = 0.5693 b) sin x = –0.4792

c) cos x = –0.428 d) cos x = 0.9283

P 27.5o

25 m 18 m

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The sine rule and cosine rule. Find the length x in the following triangles:

Mixed problems involving sine rule, cosine rule and Pythagoras’ Theorem in two and three dimensions.

Finding the area of a triangle using ½ ab Sin C

x 12 cm 54

o

14 cm x

58o

56.4 m 78o

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1. a) State Pythagoras’ Theorem.

b) Which is the longest side in a right angled triangle?

2. Calculate the distance ‘z’ in each of the following triangles. Round the answers to three

significant figures where appropriate.

a) b) c)

d) e) f)

### Q

Normally we do not give you any clues in the question section, but there is one point that is so important we want to include it here at the beginning of this section.

It is this: When solving right angled triangles (that is, when you are trying to find angles or sides given some information about the triangle) you only have two sets of tools available – Pythagoras’ Theorem

and Trigonometry – there is nothing else.

If you have this type of question in the examination, which you are almost certain to have, don’t sit there scratching your head. Decide whether to use Pythagoras or Trigonometry and get on with the question. How do you decide? Easy! If the question does not involve angles – you are not given angles and you are not asked to find angles given the three sides of a triangle – it must be Pythagoras’ Theorem. If you are given angles to use or are asked to find an angle, you should use Trigonometry. Simple as that!

In some more complicated examples, you may be given angles that you don’t need to use straight away (they are ready for a later part of the question) as well as the lengths of sides, but by the time you get to those you will, hopefully, have had plenty of practice and will be able to pick your way through the

questions.

z

3 m 4 m

z

5 cm

6 cm 12.9 m

11.8 m

z

z

12 m 13 m

z

6.33 mm

8.5 mm

z 3.2 cm

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3. A ladder is 15 m long and leans against a wall. The foot of the ladder is 4 m from the bottom of the wall. How far up the wall does the ladder reach?

If the ladder is moved so that its foot is only 3 m from the base, how much further up the wall will the ladder reach?

4. What is the length of a diagonal of a rectangle 16.2 Km by 5.7 Km ?

5. A ship left a port and travelled 152 Km due east and then 193 Km due north. How far was it

from its starting point?

6. A flagpole is 22 m high and is constrained from bending in high winds by cables each 31 m

long fixed into the ground. Each cable is stretched tight.

How far is each cable fixed into the ground from the foot of the flagpole?

7. Two points A and B, are drawn on a co-ordinate grid. A is (–2, 5) and B is (7, –9). What is

the distance between A and B?

8. The top of a chimney on one house is 9.3 metres above the ground. The top of a chimney on

another house is 6.9 metres above the ground. The chimneys are 15.6 metres apart, measured horizontally.

A bird flies directly from the top of the higher chimney pot to the lower. How far does it fly?

9. Two points A and B, are drawn on a co-ordinate grid. A is (a, b) and B is (c, d). What is

the distance between A and B?

10. Two points A and B, are drawn on a co-ordinate grid. A is (x, y) and B is (x + a2 – 1, y + 2a).

What is the distance between A and B?

11. Show that a triangle with sides 3 367, 3 456 and 4 825 units respectively is right angled.

12. Show that a triangle with sides of length 2pq, p2 – q2 and p2 + q2 will always be a right

angled triangle, provided p and q are positive numbers and p>q.

13. What is the height of the vertical line of symmetry of this isosceles triangle?

14. What is the area of each of these isosceles triangles?

a) b) c)

11 cm

9 cm 11 cm

15 cm 12 cm 12 cm

10.2 m 8.4 m

8.4 m 6.7 mm 4.3 mm

(8)

15. A ship sails from a port P to an island Q, 100 Km south, then on to another island R, 142 Km east of Q. From R it travels south again 110 Km to another island S and finally east a distance of 205 Km to its destination T.

a) How far is it from P to R as the crow flies?

b) How far is it from R to T as the crow flies?

c) How far is it from P to T as the crow flies?

16. Calculate the length of the diagonal from A to G in this cuboid.

B

17.

Calculate:

a) the distance OB

b) the vertical height of the pyramid OA

c) the distance AM, where M is the midpoint of BE

d) the total surface area of the pyramid

e) the volume of the pyramid.

18.

A C

D

E

F

H

G 8.5 cm

7.6 cm

7.2 cm

A 12 cm

C B

O M

D 8 cm

E

The diagram shows a square based right pyramid. The length of the sides of the base is 8 cm and the length of the sloping edges is 12 cm.

X

Z

Y

A line is drawn ‘diagonally’ on a can from X to Z via Y, as shown in the diagram. (Z is directly below X.) The can has a diameter of 13 cm and a height of 15 cm.

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19. Find the value of the following, correcting to three significant figures where appropriate: a) sin 57o b) sin 0o c) sin 90o d) sin 69o e) sin 137o

f) cos 26o g) cos 90o h) cos 0o i) cos 48o j) cos 141o k) tan 35o l) tan 45o m) tan 12o n) tan 30o o) tan 127o

20. Find the value of the following, correcting to three significant figures where appropriate:

a) 16.8 sin 12o b) 9 tan 37o c) 56 d) 19.4 e) 45.6 sin 105o

cos 18o sin 69o

f) 7 cos 88o g) 45 h) cos 56.4o i) 19.5 cos 72o j) 89.56

tan 36.8o sin 23.5o cos 141o

21. Find the value of sin 63o divided by cos 63o . Find the value of tan 63o.

22. Find the value of (sin 37.5o)2 + (cos 37.5o)2 .

23. Find the value of (sin 16.83o)2 + (cos 16.83o)2 .

24. Find the value of (sin 256o)2 + (cos 256o)2 .

25. Find the value of x in each of the following, giving the answers correct to one decimal place:

a) sin x = 0.7382 b) cos x = 0.5915 c) tan x = 0.3347 d) sin x/cos x = 0.2142

26. For each of the following triangles, find the value of sin x, cos x and tan x.

40 cm

a) b) c) 5 cm 26 cm 10 cm

41 cm 3 cm

24 cm 9 cm

4 cm

sin x = sin x = sin x = cos x = cos x = cos x = tan x = tan x = tan x =

x

x

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27. In each of the following triangles, find the length of the side x. a) b) c)

28. In each of the following triangles, find the length of the side x. a) b) c)

29. What are the values of the angles in a 3-4-5 triangle?

30. What are the values of the angles in a 5-12-13 triangle?

31. θ is one angle in a right angled triangle. If cos θ is 0.7352, what is the value of θ ?

32. θ is one angle in a right angled triangle. Sin θ = 9/41 and the opposite side to θ is 27 m

long. What is the length of the other two sides?

33. θ is one angle in a right angled triangle. Cos θ = 12/13 and the hypotenuse is 65 cm

long. What is the length of the other two sides?

34. θ is one angle in a right angled triangle. Tan θ = 48/55 and the hypotenuse is 438 cm

long. What is the length of the side adjacent to θ ?

35. Using the triangle below, give an algebraic proof that the sine of an angle divided by its

cosine is equal to the tangent of the angle.

36. Using the same triangle as in question 35, show that the square of the sine of an angle plus the square of the cosine of the same angle is 1.

x 5 cm 340

x

430 4 cm

510 5 cm

x

x 8.7 cm

510

x 38.60 12.9 cm

67.30 7 cm

x

r p

q

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37.

Here is a neat proof of Pythagoras’ Theorem that shows the close relationship between this theorem and trigonometry. Answer the parts of the question carefully to complete the proof.

a) In triangle ACT, cos A = Use the small letters and write the answer as a fraction.

b) In triangle ABC, cos A = Use the small letters and write the answer as a fraction.

c) Use your answers to parts a) and b) to complete the equation b2 = .

d) In triangle BCT, cos B = Use the small letters and write the answer as a fraction.

e) In triangle ABC, cos B = Use the small letters and write the answer as a fraction.

f) Use your answers to parts d) and e) to complete the equation a2 = .

g) Use your answers to parts c) and f) to complete the equation a2 + b2 = .

h) Hence complete the proof of Pythagoras’ Theorem.

38. A tower is 43m high. Julia stands a certain distance from the tower and measures the angle

of elevation from the ground to the top of the tower to be 18o. How far is she from the

tower?

39. Michael is flying a kite on a string 26 metres long. The angle of elevation of the kite is 42o.

Michael’s hand is 1.4 metres above the ground. How high is the kite above the ground?

40. A rectangle ABCD has dimensions AB = 8 cm and BC = 14 cm. What is the angle ACB ?

41. A cliff is 156 metres high. A boat is 720 metres from the base of the cliff. What is the angle

of depression of the boat from the top of the cliff?

42. If Larry stands 65 m from a building and the angle of elevation to the top of the building

from the ground is 47o, how high is the building?

C

T

B A

b a

m n

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43. A section of straight road 1.5 Km long rises 75 metres over its length. What is the average angle that the road makes with the horizontal?

44. Which bearings are represented by the following compass directions:

a) East b) West c) South d) North e) North East f) South West

45. A ship travels 135 Km on a bearing of 135o. How far is it east of its starting point?

46. An aircraft flies 340 Km on a bearing of 285o. How far is it west of its starting point and how

far is it north of its starting point?

47. A mountain P is 45 Km east and 34 Km north of another mountain Q. What is the bearing of

mountain Q from mountain P ? What is the back bearing of mountain P from mountain Q ?

48. The angle of elevation to the top of a mountain from the base camp of a mountaineering

expedition is measured very accurately to be 3.582o. The horizontal distance from the base

camp to the mountain is 56.43 Km. How high is the mountain above the base camp?

49. A ship sails from a port P on a bearing of 206o for a distance of 56 Km to an island Q and

then changes direction to a bearing of 143o for a distance of 98 Km to another port R. How

far is the ship now east of its starting point and how far is it south of its starting point? Give each distance correct to the nearest 100 metres.

50. An aircraft flies 890 Km from an airport on a bearing of 340o and then changes to a bearing

of 203o and flies another 760 Km.

a) How far is the plane now west of the airport?

b) Is the plane now north or south of the airport and by how much?

c) What bearing must the plane now fly in order to return directly to the airport?

d) How far is the return flight assuming a direct return?

51. Michelle measures the angle of elevation to the top of a tower from the ground to be 63o.

Peter stands 22 metres behind her and measures the angle of elevation to the top of the

tower to be 35o.

a) How far is Michelle from the tower?

b) How tall is the tower?

52. A point P and two flagpoles are in a straight line. P is positioned such that P and the tops of

the flagpoles are also in a straight line. The flagpoles are 25 metres apart, the smaller is

18 metres high and the angle of elevation from P to the tops of the flagpoles is 27.5o.

Find the height of the taller flagpole. P 27.5o

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53. Find the distance d in the diagram.

54.

The diagram represents a rectangular playing field PQRS. FR is a floodlight gantry in one corner of the field.

PS = 98 metres. RS = 55 metres. The angle of elevation of the floodlight from the point Q is 19o.

Find:

a) The height of the floodlight gantry.

b) The length of the diagonal PR.

c) The angle of elevation of the floodlight F from point P.

d) The distance FS.

e) The angle of elevation of the floodlight from the point S.

55. The base ABCD of a right square based pyramid is laid horizontally on a table. The

vertex P is directly over the centre of the base. AB = 12 cm. AP = 22 cm.

Find:

a) The height of the pyramid.

b) The angle PDB.

c) The area of one of the sloping faces.

39o 42o

23 m

d

F

P

S

R Q 19o

55 m 98 m

(14)

56.

In the diagram, the radius of the circle is 9.83 metres, the angle ABC is 38.5o.

Find:

a) The length of OB.

b) The area of OABC.

57. Give the angles, x, between 0o and 360o for each of the following:

a) sin x = 0.5693 b) sin x = –0.4792 c) cos x = –0.428 d) cos x = 0.9283

e) tan x = 6.723 f) tan x = –0.2735 g) sin x = 2.452 h) 1/

cos x = –6.32

58. State the sine rule for a triangle ABC.

59. Find the length x in the following triangles:

a) b)

c) d)

O A

C

B

x 58o

56.4 m 78o

38.6o

x

19 cm 48.2o

27 m 126o

x

32.9o x

89.4 mm 22o

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60. A triangle has vertices A, B and C.

a) Complete this form of the cosine rule:

a2 =

b) Complete this form of the cosine rule:

Cos A =

61. Find the length x in the following triangles:

a) b)

c) d)

62. Find the angle marked P in the following triangles: a) b)

x

12 cm 54 14 cm

o

29.5o x

9.3 m

10.9 m

198 mm 157o

88 cm 25.8o

38 cm

x x

183 mm

P Q

R 12 m

19 m 18 m

P

Q

9.365 Km

5.942 Km

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63. What happens when you try to use the cosine rule to find angle Q in this triangle?

64. Explain how, given a triangle with no right angles, you would know whether to use the sine

rule or the cosine rule, then try the questions below.

65. Peter is standing some way from the base of a small hill. On top of the hill is a vertical

tower.

Peter measures the angle of elevation to the top of the tower to be 29o and the angle to the

bottom of the tower to be 18o. The direct distance of Peter from the base of the tower is

134 metres.

Draw a diagram, labelling Peter’s position P, the base of the tower B and the top of the tower T.

a) Find the angle TPB.

b) Find the angle PTB.

c) Find the height of the tower.

66. A plane flies on a bearing of 067o for 75 Km and then on a bearing 112o for a further 99 Km.

Find the distance, as the crow flies, of the plane from its starting position.

67. A trapezium PQRS has <QPS = 107o, PQ = 84.5 m, PS = 76.4 m and SR = 115.6 m.

a) Find the length of the diagonal QS.

b) Find angle QSP.

c) Find angle QSR.

d) Find the length of QR.

e) Find the area of the trapezium.

P 4.2 cm

3.8 cm

Q

8.1 cm

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68. A surveyor is measuring a triangular piece of land TUV. TU is 56 m and TV is 65 m.

The angle UTV is 69o.

Calculate:

a) The perimeter of the triangle TUV.

b) The area of the triangle.

69. A triangle ABC has AB = 10 cm, BC = 6 cm and angle BAC = 31o.

Show that there are two possible values for angle C.

Demonstrate this difference by sketching and labelling the two possible triangles.

70. STU is a right angled triangle with the right angle at T.

P is a point somewhere on TU. Angle TPS = 37o, angle TUS = 21o and PU = 243 metres.

Find the length of ST. 71.

The diagram shows a structure made from steel struts that is to be used as part of a new bridge construction. It is assembled from two rectangular frames, PSUR and RUTQ joined at right angles along the line RU. Additional struts PQ and ST are welded into position.

P

S

U R

The dimensions are shown on the diagram. T

Q

Calculate the following:

a) The length of PQ to 2 d.p. b) The distance PT to 2 d.p. c) The length QU to 2 d.p. d) <PQR e) <UQS

72. Three points X, Y and Z lie on a circle. The angle YXZ is 42o, the angle XYZ is 57o and the

distance YZ is 300 mm.

a) Calculate the length of XY.

b) Hence, or otherwise, find the area of triangle XYZ.

c) Find the radius of the circle.

73. A triangle XYZ has XY = 54 cm, YZ = 71 cm and <XYZ = 4α

a) Find the area of the triangle in terms of sin 4α .

b) What value of α gives the maximum area of the triangle?

c) What is the maximum area of the triangle?

d) There are two possible values of α that give an area of 1200 cm2. What are these two

possible values?

7.84 m

20.74 m

References

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