Chapter 5 Trigonometry test questions

Student name:

1 Find, correct to four decimal places:

a sin 71° b cos 54° c tan 112°13′
d sin 13°48′
2 Find the acute angle 𝜃, correct to the nearest minute, given that:
a sin 𝜃 = 0.75 b cos 𝜃 = 0.45 c tan 𝜃 = 1.17
d tan 𝜃 = −2.12
3 Find, correct to two decimal places, the side marked 𝑥 in each triangle below.
a b

c d

4 Find, correct to the nearest minute, the angle 𝜃 in each triangle below.
a b

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 c d

5 Use the special triangles to find the exact values of:

a sin 60 b tan 45 c cos 45
d sec 60 e cosec 45 f cot 30
6 A vertical pole stands on level ground. From a point on the ground 7 metres from its
base, the angle of elevation of the top of the pole is 29°. Find the height of the pole,
correct to the nearest centimetre.
7 At what angle, correct to the nearest degree, is a 5 metre ladder inclined to the
ground if its foot is 2.75 metres out from the wall?
8 A motorist drove 60 km from town A to town B on a bearing of 054°T, and then
drove 85 km from town B to town C on a bearing of 144°T
a Explain why ∠ABC = 90°
b How far apart are the towns A and C, correct to the nearest kilometre?
c Find ∠BAC, and hence find the bearing of town C from town A, correct to the
nearest degree.
9 Sketch each graph for 0° ≤ 𝑥 ≤ 360°,
a 𝑦 = cos 𝑥 b 𝑦 = sin 𝑥 c 𝑦 = tan 𝑥
10 Write each trigonometric ratio as the ratio of its related acute angle, with the
correct sign attached.
a sin 135° b cos 300° c tan 225 °
d cos 240°
11 Find the exact value of:
a cos 150 ° b sin 300° c tan 240°
d cos 240°
12 Use the graphs of the trigonometric functions to find these values, if they exist
a sin 180° b tan 270° c cos 0°
d tan 300°

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13 Use Pythagoras’ theorem to find whichever of x, y or r is unknown. Then write down
the values of sin 𝜃, cos 𝜃 and tan 𝜃
a b

13
14 a If tan α = and 𝛼 is acute, find the values of sin 𝛼 and cos 𝛼.
9

√7
b If sin 𝛽 = and 𝛽 is acute, find the values of cos 𝛽 and tan 𝛽.
3
43
c If tan 𝜃 = − 13 and 270° < 𝜃 < 360°, find the values of sin 𝜃 and cos 𝜃.

√7
d If sin 𝐴 = and 90° < 𝐴 < 180°, find the values of cos 𝐴 and tan 𝐴.
6

15 Simplify (all trigonometric functions do not equal zero):

1 1 sin 𝜃
a b c
sec 𝜃 cot 𝜃 tan 𝜃

d 1 − cos2 𝜃 e cosec 2 𝜃 − cot 2 𝜃 f tan2 𝜃 − sec 2 𝜃

16 Prove the following trigonometric identities.
a sin 𝜃 cosec 𝜃 = 1 b cot 𝜃 tan 𝜃 = 1
sin 𝜃
c = cos 𝜃 d 3 sin2 𝜃 − 2 = 1 − 3 cos 2 𝜃
tan 𝜃
cot 𝜃 sec 𝜃 cos 𝜃+sin 𝜃 cosec 𝜃 tan 𝜃
e + tan 𝜃 = f sin2 𝜃 cot 2 𝜃 − = − sin2 𝜃
sin 𝜃 sin2 𝜃 sec 𝜃

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17 Solve each trigonometric equation for 0° ≤ 𝑥 ≤ 360°.
1 1
a sin 𝑥 = 2 b cos 𝑥 = c tan 𝑥 = 1
√2
1 2
d sin 𝑥 = 0 e tan 𝑥 = 2 f sec 𝑥 =
√3

g √3 sin 𝑥 − 1 = 0 h 2cos 𝑥 − √3 = 0 i tan2 𝑥 = 1

1 √3
j sin 2𝑥 = 2 k cos(𝑥 − 60°) = l cos 𝑥 = −√3 sin 𝑥
2

18 Solve each equation for 0° ≤ 𝑥 ≤ 360° by reducing it to a quadratic equation in u.

Give your solution correct to the nearest minute where necessary.
a cos 𝜃 − 2 cos2 𝜃 = 0 b 2 sin2 𝜃 + 3 sin 𝜃 = 1
c 3 tan2 𝜃 − 7 tan 𝜃 = 4
19 Use the sine rule or the cosine rule in each triangle to find 𝑥, correct to one decimal
place.
a b

c d

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20 Calculate the area of each triangle, correct to the nearest cm2 .
a b

21 Use the sine rule or cosine rule in each triangle to find 𝜃, correct to the nearest
minute.
a b

c d

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22 A triangle has sides 8 cm, 9 cm and 11 cm. Use the cosine rule to find one of its
angles and hence find the area of the triangle, correct to the nearest cm2.
23 a Find the side A in ∆𝐴𝐵𝐶, where ∠𝐶 = 60°, 𝑏 = 24 cm and the area is 30 cm2.
b Find the size of ∠𝐵 in ∆𝐴𝐵𝐶, where 𝑎 = 9 cm, 𝑐 = 8 cm and the area is 18 cm2.
24 A helicopter H is hovering above a straight, horizontal
road AB of length 750 m. The angles of elevation of H
from A and B are 56o and 38o respectively. The point C
lies on the road directly below H.
750 sin 56°
a Use the sine rule to show that 𝐻𝐵 = .
sin 86°

b Hence find the height CH of the helicopter above to road, correct to the nearest
metre.
25 A man is sitting in a boat at P, where the angle of
elevation of the top T of a vertical cliff BT is 13°. He
then rows 50 metres directly towards the cliff to Q,
where the angle of elevation of T is 27o.
50 sin 13
a Show that 𝑇𝑄 = using the sine rule.
sin 14

b Hence find the height h of the cliff, correct to the nearest tenth of a metre.
26 A ship sailed 120 nautical miles from port P to port Q on a
bearing of 040oT. It then sailed 250 nautical miles from
port Q to port R on a bearing of 140oT.
a Explain why ∠𝑃𝑄𝑅 = 80°.
b Find the distance between ports R and P, correct to the
nearest nautical mile.
c Find the bearing of port R from port P, correct to the nearest degree.
27 From two points P and Q on horizontal ground, the angles
of elevation of the top T of an 8 m monument are 15° and
12° respectively. It is known that ∠𝑃𝐵𝑄 = 70°, where B is
the base of the monument.
a Show that 𝑃𝐵 = 8 tan 75°, and find a similar expression
for QB.
b Hence determine the distance between P and Q, correct
to the nearest metre.
28 The diagram below shows an open wooden crate in the shape of a rectangular
prism. The base is 1.7 metres by 1.1 metres, and the height is 0.8 metres.

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 a Find, correct to the nearest millimetre the length of the base diagonal BD.

b Find, correct to the nearest millimetre the length of the longest metal rod BS will
fit in the box.
c Find, correct to the nearest minute angle that the rod BS makes with the base.
29 The points P, Q and B lie in a horizontal plane. From P, which is due west of B, the
angle of elevation of the top of a tower AB of height h metres is 36o. From Q, which
is on a bearing of 196o from the tower, the angle of elevation of the top of the tower
is 27o. The distance PQ is 250 metres.
a Explain why ∠𝑃𝐵𝑄 = 74°.
2502
b Show that ℎ2 = cot2 36°+cot2 27°−2 cot 27° cot 36° cos 74°

c Hence find the height of the tower, correct to the

nearest metre.

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Chapter 5 Trigonometry test answers

1 a 0.9455
b 0.5878
c -2.4484
d 0.2385
2 a 48o35’
b 63o15’
c 49o29’
d -64o45’
3 a 15.05
b 8.82
c 12.31
d 5.08
4 a 37o59’
b 61o21’
c 51o1’
d 49o4’
√3
5 a 2

b 1
1
c
√2

d 2

e √2

f √3
6 3.88 metres
7 56.63 metres
8 b 104
c 55o, 109o

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9 a

10 a sin 45
b cos 60
c tan 45
d -cos 60
√3
11 a − 2

√3
b − 2

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 c √3
1
d −2

12 a 0
b Does not exist
c 1

d −√3
3 4 3
13 a sin 𝜃 = 5, cos 𝜃 = 5 and tan 𝜃 = 4

7 √6 7
b sin 𝜃 = , cos 𝜃 = and tan 𝜃 =
√55 √55 √6
13 9
14 a sin 𝛼 = 5√10, cos 𝛼 = 5√10

√2 7
b cos 𝛽 = , tan 𝛽 = √2
3

43 13
c sin 𝜃 = − , cos 𝜃 =
√2018 √2018

√29 7
d cos 𝐴 = − , tan 𝐴 = −√29
6

15 a cos 𝜃
b tan 𝜃
c cos 𝜃
d sin2 𝜃
e 1
f -1
17 a 30°, 150°
b 45°, 315°
c 45°, 225°
d 0°, 180o, 360°
e 26o34’, 206o34’
f 30°, 330°
g 35°16’, 144°44′
h 30°, 330°
i 45°, 135°, 225°, 315°

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 j 15°, 75°, 195°, 255°
k 30°, 90°
l 150°, 330°
18 a 60°, 90°, 270°, 300°
b 0°, 30°, 150°, 360°
c 53°8′, 74°45′ , 233°8′, 254°45′
19 a 13.5
b 4.5
c 15.1
d 7.5
20 a 48 cm2
b 58 cm2
21 a 38°37’
b 63°45’
c 33°58’
d 118°10’
22 35 cm2
23 a 2.89 cm
b 30°
24 b 384 m
25 b 21.1 m
26 b 258 nm
c 113
27 b 39 m
28 a 2.025 m
b 2.177 m
c 21°33’
29 c 121 m

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