236 1 Sketch each triangle. Label the hypotenuse (hyp), opposite

1 Sketch each triangle. Label the hypotenuse (hyp), opposite (opp) and adjacent
(adj) sides in relation to the marked angle x.
a
b
c
x
x
x
e x
d
x
f
x
2 This question is about right-angled triangles.
a Draw accurately five different right-angled triangles
that each have a 30° angle.
hyp
opp
30°
Label the opposite and hypotenuse of each triangle.
b Copy and complete this table by measuring the opposite
and hypotenuse on each triangle correct to the nearest millimetre.
c What do you notice
about all the values in
the final column?
Compare your results
with your neighbour.
Triangle Opp (cm) Hyp (cm)
1
2
3
4
236
5
Opp ÷ Hyp
Geometry and measures GM4.4 Trigonometry
3 Use your calculator to find these. Give your answers correct to three significant
figures where necessary.
a sin 40°
b cos 60°
c tan 45°
d cos 63°
e tan 59°
f sin 82°
g tan 12.5°
h cos 35.1°
i sin 9.76°
j cos 78.4°
4 Look at each triangle.
Decide which two sides have been labelled. Write whether sin, cos or tan should
be used.
a
27°
b
x
c
x
17 cm
30°
15 cm
7.3 cm
x
48°
d
e
8 cm
f
54°
x
x
x
57°
35°
13.8 cm
12.3 cm
5 For all the triangles in question 4, work out the length of each side marked x.
Give your answers correct to one decimal place.
237
Geometry and measures GM4.4 Trigonometry
6 Work out the length of each marked side. Give your answers correct to
three significant figures.
a
9 cm
b
30°
c
14.8 cm
a
b
23 cm
50°
62°
d
e
f
e
63°
59°
8.5 cm
36 cm
33°
d
f
6.8 cm
g
8.6 cm
71°
h
g
5.4 cm 68°
i
9.3 cm
23°
i
h
7 A ladder stands on horizontal ground and leans against a
vertical wall. The ladder is 3.5 m long and makes an angle of
32° with the wall. How far up the wall does the ladder reach?
Give your answer correct to the nearest centimetre.
3.5 m
32°
8 From a point 30 m from the foot of a building, the
angle of elevation to the top of the building is 63°.
Work out the height of the building.
Give your answer correct to the nearest metre.
238
63°
30 m
c
Geometry and measures GM4.4 Trigonometry
9 A ski run is 1750 m long and slopes at an angle of 23° to the horizontal.
A skier skis down the complete ski run. How far will the skier descend
vertically in height? Give your answer correct to the nearest metre.
10 Work out the height of the isosceles triangle.
16 cm
Give your answer correct to one decimal place.
16 cm
55°
55°
11 In each of the questions, find the value of x.
Give your answers correct to three significant figures.
a cos x = 0.5
b tan x = 1
c sin x = 0.866
d tan x = 1.5
e sin x = 0.32
f cos x = 0.768
g cos x = 0.128
h tan x = 6.31
12 In each triangle, work out the size of the angle marked x.
Give your answers correct to one decimal place.
a
b
7.1 cm
17 cm
x
c
7.2 cm
x
x
5.8 cm
9.1 cm
8 cm
d
e
f
12 cm
x
8 cm
5.6 cm
13.8 cm
x
21.6 cm
x
10.3 cm
13 A ladder of length 3.5 m rests against a vertical wall so that the base of the
ladder is on horizontal ground 2 m away from the wall. Calculate the angle
between the ladder and the wall. Give your answer correct to one decimal
place.
239
Geometry and measures GM4.4 Trigonometry
14 A rectangle has length 15 cm and width 9 cm. Work out the angle between any
diagonal and the longest side of the rectangle.
Give your answer correct to one decimal place.
15 A boy lies on the top of a 67 m vertical cliff.
He sees a boat that is 120 m away from the base of the cliff.
Work out the angle of elevation of the boy from the boat.
Give your answer correct to one decimal place.
16 In each triangle, work out the size of the angle or the side marked x.
Give your answers correct to three significant figures.
a
b
c
7.3 cm
10.2 cm
x
10.6 cm
9.7 cm
6.8 cm
x
x
73°
d
e
f
40°
x
28.5 cm
8.4 cm
12.4 cm
x
x
39°
5.1 cm
g
h
26 cm
x
71°
16.8 cm
240
i
24 cm
19°
10 cm
7.2 cm
x
x
Geometry and measures GM4.4 Trigonometry
17 Work out the area of the trapezium.
8 cm
Give your answer correct to three significant
figures.
40°
11 cm
18 An isosceles triangle has sides of 8 cm, 8 cm and 10 cm.
a Work out the sizes of all the angles in the triangle.
b Work out the height of the triangle.
19 Work out the length of each marked side. Give your answers correct to
one decimal place.
a
x
b
21°
6.3 cm
c
21°
8.7 cm
x
x
44°
5 cm
d
e
x
35°
f
x
17.2 cm
48°
11 cm
x
60°
23.4 cm
20 An isosceles triangle has two equal angles of 50°
and a height of 7.8 cm. Work out the length of all
the sides of the isosceles triangle. Give your answers
correct to one decimal place.
7.8 cm
50°
50°
241
Geometry and measures GM4.4 Trigonometry
21 Work out the length of each marked side. Give your answers correct to
one decimal place.
a
b
x
c
x
29°
27 cm
27 cm
40°
34°
14.6 cm
x
d
e
10 cm
f
7.5 cm
19.8 cm
x
74°
x
65°
x
59°
22 An escalator is inclined at an angle of 23° to the horizontal.
The vertical distance that the escalator travels through is 8.5 metres.
How long is the escalator? Give your answer correct to the nearest 10 cm.
23 The diagram shows a quadrilateral.
C
a Work out the length of AC.
29°
Give your answer correct to 1 d.p.
55°
b Work out the length of AD.
Give your answer correct to 1 d.p.
B
7 cm
24 Work out the size of the angle marked x.
A
x
Give your answer correct to one decimal place.
11 cm
6 cm
35°
242
D

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