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Form 2 Mathematics Trigonometry 1 exam questions and answers
Given that
3
θ
is an acute angle and sin
3
θ
= Cos
2
θ
, find the value of
θ
.
(1m 11s)
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1.
When the angle of elevation of the sun is
58
0
, a vertical pole casts a shadow of length 5 m on a horizontal ground. Find the height of the pole.
2.
The angle elevation of the top of a cliff from point P is
45
0
. From a point Q which is 10 m from P towards the foot of the cliff, the angle of elevation is
48
0
. Calculate the height of the cliff.
3.
Towns A, B, C and D are such that A is 15 km north of B, C is 8 km east of B, D is directly east of A and on a bearing
060
0
from C. Find the distance between towns A and D.
4.
A flag post 10 m long is fixed on top of a tower. From a point on horizontal ground, the angles of elevation of the top and bottom of the flag post are
40
0
and
33
0
respectively. Calculate (a) The height of the tower. (b) The shortest distance from the point on the ground to the top of the flag post.
5.
A man walks directly from a point A towards the foot of a tall building 240 m away. After covering 180 m, he observes that the angle of elevation of the top of the building is
45
0
. Determine the angle of elevation of the top of the building from A.
6.
There are two sign-posts A and B on the edge of a road. A is 400 m to the west of B. A tree is on a bearing of
060
0
from A and on a bearing of
330
0
from B. Calculate the shortest distance of the tree from the edge of the road.
7.
A point A is directly below a window. Another point B is 15 m from A at the same horizontal level. From B the angle of elevation of the bottom of the window is
30
0
and the angle of elevation of the top of the window is
35
0
. Calculate the vertical distance (a) From A to the top of the window (b) From A to the bottom of the window (c) From the bottom to the top of the window
8.
In the figure below angle ABC =
30
0
, angle ACB = angle ADC =
90
0
, AD =
4
3
√
3
cm and DC = 4 cm. Calculate the length of (a) AC (b) BC
9.
Two straight paths are perpendicular to each other at a point P. One path meets a straight road at point A while the other meets the same road at B. Given that PA is 50 metres while PB is 60 metres, calculate the obtuse angle made by PB and the road.
10.
Given that sin
(
90
-
x
)
0
= 0.8, where x is an acute angle, find without using Mathematical Tables the value of tan
x
0
.
11.
The diagram below represents a school gate with double shutters. The shutters are each opened through an angle of
63
0
. The edges of the gate, PQ and RS are each 1.8 m. Calculate the shortest distance QS, correct to 4 significant figures.
12.
Given that x is an acute angle and cos x =
2
√
5
5
, find without using mathematical tables or a calculator, tan
(
90
-
x
)
0
.
13.
Points L and M are equidistant from another point K. The bearing of L from K is
330
0
. The bearing of M from K is
220
0
. Calculate the bearing of M from L
14.
The diagram below represents two vertical watch-towers AB and CD on a level ground. P and Q are two points on a straight road BD. The height of the tower AB is 20 m and road BD is 200 m. (a) A car moves from B towards D. At point P, the angle of depression of the car from point A is
11.3
0
. Calculate the distance BP to 4 significant figures. (b) If the car takes 5 seconds to move from P t
15.
An electric pole is supported to stand vertically on a level ground by a tight wire. The wire is pegged at a distance of 6 metres from the foot of the pole as shown. The angle which the wire makes with the ground is three times the angle it makes with the pole. Calculate the length of the wire to the nearest centimeter.
16.
Given that
3
θ
is an acute angle and sin
3
θ
= Cos
2
θ
, find the value of
θ
.
17.
Given that sin
(
x
+
60
)
0
=
cos
(
2
x
)
0
, find tan
(
x
+
60
)
0
18.
Given that
tan
x
0
=
3
7
, find
cos
(
90
-
x
)
0
giving the answer to 4 significant figures.
19.
A piece of wire is bent into the shape of an isosceles triangle. The base angles are each
48
0
and the perpendicular height to the base is 6 cm. Calculate, correct to one decimal place, the length of the wire.
20.
Solve the equation
sin
(
1
2
x
-
30
0
)
= cos x for
0
<
x
<
90
0
.
21.
Given that
sin
2
x
=
cos
(
3
x
-
10
0
)
find tan x, correct to 4 significant figures.
22.
Express each of the following in degrees and minutes: (a) 15.3° (b) 25.75° (c) 30
1
2
° (d) 34
3
4
°
23.
Read from tables the tangent of the following: (a) 84° (b) 43° 51' (c) 57.17°
24.
Find from the tables the angle whose tangent is: (a) 0.3317 (b) 0.6255 (c) 1.6391 (d) 0.4444 (e) 0.0122
25.
Find from the tables the angle whose tangent is: (a) 0.8799 (b) 0.1867 (c) 0.5903 (d) 5.1006 (e) 1.0000
26.
Find the cosine and sine of each in the following marked angles. (Units are in centimetres)
27.
A boy on top of a vertical wall 13.5 m high throws a ball down and notices that the ball hits a stone on the ground 18 m away from the foot of the wall. Calculate the angle of depression of the stone from the top of the wall.
28.
A ladder leans against a wall so that its foot is 2.5 m away from the foot of the wall and its top is 4 m up the wall. Calculate the angle it makes with the ground.
29.
A pedestrian notices that a tower is at a horizontal distance of 30 m away from him and that the angle of elevation of the top of the tower from where he is, is 35°. Find the height of the tower.
30.
A tree casts a shadow 20 m long. Find the height of the tree if the angle of elevation of the top of the tree from the tip of the shadow is 31°.
31.
In a right-angled triangle, the shorter sides are 4.5 cm and 9.2 cm long. Find the sizes of its acute angles.
32.
One of the diagonals of a rhombus is 28 cm long and one of its angles is 70°. Calculate the length of the second diagonal and hence the side of the rhombus (two possible answers).
33.
From a window 25 m above a street, the angle of elevation of the top of a wall on the opposite side is 15°. If the angle of depression of the base of the wall from the window is 35°, find: (a) the width of the street. (b) the height of the wall on the opposite side.
34.
An aircraft flying into an airport calls out the control tower and says it is at height of 500 m above the tower. If its horizontal distance from the tower is 8 km, calculate its angle of elevation from the top of the tower.
35.
ABC is a right-angled triangle inscribed in a circle ABC with AC as the diameter. See the figure below. If BC = 25 cm and AB = 40 cm, calculate angles BAC and ACB.
36.
The figures below shows five towns A, B, C, D and X connected by straight roads. BX = 40 km and X D = 48 km. If
∠
AXB = 56°, calculate the distances AB and BC.
37.
Calculate b, x, c and y in centimetres if a = 24 cm.
38.
The figure below shows a right-angled triangle ABC. AD is perpendicular to BC. If
∠
ABC = 30° and AD = 6.5 cm, find the sides marked x, y, p and q.
39.
(a) Find by scale drawing the angle whose sine is: (i)
7
12
(ii) 0.5 (iii) 0.30 (iv) 0.6 (v) 0.7
40.
(a) If sin
θ
=
3
5
find: (i) cos
θ
(ii) tan
θ
(b) If sin
θ
=
1
2
, find tan
θ
. (c) If tan
θ
=
1
2
√
3
find cos
θ
. (d) If cos
θ
=
1
2
√
2
, find sin
θ
.
41.
Find from tables the angle whose sine is: (a) 0.3367 (b) 0.5871 (c) 0.0523 (d) 0.8500 (e) 0.1822 (f) 0.9834 (g) 0.5012 (h) 0.2518
42.
Find from the tables the angle whose cosine is: (a) 0.1643 (b) 0.7196 (c) 0.9970 (d) 0.8660 (e) 0.4009 (f) 0.9481
43.
Read from the tables the sine of: (a) 31.46° (b) 77° 34' (c) 52° 9' (d) 66° 31'
44.
Read from the tables the sine of: (a) 6.76° (b) 40.13° (c) 26° 47' (d) 13.07°
45.
Read from the tables the cosine of: (a) 79° 42' (b) 24.23° (c) 5° 37' (d) 60° (e) 88° 59' (f) 55.97° (g) 33.33° (h) 17° 52'
46.
PQRS is a rhombus of side 7 cm and
∠
PQR is 65°. Calculate the lengths of PR and QS.
47.
Find the values of the unknown side in each of the following figures:
48.
A ladder 10 m long leans against a wall as shown in the figure below. If its foot is 3 m from the wall, calculate: (a) the vertical distance from the top of the ladder to the ground. (b) the angle the ladder makes with the ground.
49.
From a point P on the ground, 20 m away from the foot of a building, the angle of elevation of the top of the building is 25°. Find: (a) the height of the building. (b) the shortest distance to the top of the building from point p.
50.
The figure below shows an isosceles triangle in which AB = AC = 6cm. Angle BAC is 80°. Calculate the length of BC.
51.
The angle of depression of a car from the top of a building 8 m high is 44°. Find the distance of the car from the foot of the building.
52.
The figure below shows a trapezium PQRS in which PQ = 7 cm, PS = 5 cm,
∠
PQR =
∠
QRS = 90° and
∠
SPQ = 125°. Calculate the lengths QR and RS.
53.
In the figure below, PT = 3.2 cm, RS = 10 cm,
∠
SPT = 51° and
∠
QPR=33°. Calculate lengths of the sides marked a, b, c, d and e.
54.
Calculate the values of p, q, r and s.
55.
The diagonals of a rectangle PQRS are 13 cm long. If
∠
SQR = 27°, find the dimensions of the rectangle.
56.
In the figure below, WXYZ is a trapezium with WZ parallel to XY. WX = 15.61 cm, XY = 32.73 cm, ZY = 9.73 cm,
∠
WXY = 26° and
∠
ZYX = 43°. Calculate the length of WZ.
57.
In the figure below, AB = 8 cm, CD = 16 cm,
∠
ACD = 70° and
∠
BAE = 50°. Calculate ED.
58.
The angle at the vertex of a pair of dividers is 54°. The tips of the arms are 12 cm apart. If the dividers are held upright, calculate: (a) the angle between the horizontal and the arms. (b) the length of each arm. (c) the height of the vertex above the horizontal.
59.
A regular octagon of side 6 cm is inscribed in a circle. Calculate the diameter of the circle.
60.
In the figure below, QS = 20 cm, TS = 7 cm,
∠
RTS = 65° and
∠
QPR =25°. Find the length of PT.
61.
In the figure below, LN = 7.6 cm,
∠
KLM = 38°,
∠
KMN = 53°. Calculate KM and KL.
62.
If A and B are complementary angles and sin A =
4
5
, find cos B.
63.
X and B are complementary angles. If sin B = 0.9975, find X.
64.
In the figure below, AB = 16 cm and AC = 20 cm. Find: (a) sin
θ
. (b) cos
θ
.
65.
A and B are complementary angles. If A =
1
2
B
, find: (a) sin A. (b) cos A.
66.
Find the acute angle x, given that cos x° = sin 2x°.
67.
Simplify the following without using tables. Note:
√
a
×
√
a = a (a) sin 30° cos 30° (b) 4 cos 45° sin 60° (c) 3 cos 30° + cos 60° (d) tan 45° + cos 45° sin 45° (e) sin 60° cos 30° + Sin 30° cos 60° (f)
cos
2
60°+
sin
2
60° (sin
θ
×
sin
θ
is written as
sin
2
θ
)
68.
An isosceles triangle is such that AB = AC = 8 cm. If the perpendicular distance from A to BC is 6 cm, find: (a) the length of BC. (b)
∠
BAC.
69.
The angle made by the arms of an upright pair of dividers and the horizontal is 30°. The vertical distance from the horizontal to the vertex is 10 cm. Find without using tables: (a) the horizontal distance between the tips of the arms. (b) the length of the arms.
70.
The angle at the vertex of a cone is 90°. If the slant height is 3
√
2 cm, find without using tables: (a) the diameter of the cone. (b) the height of the cone.
71.
Find the height of an equilateral triangle of side x cm. Use the triangle to show that
sin
2
60° +
cos
2
60 = 1 (without using tables).
72.
In the figure below, ABC is an isosceles triangle.
∠
BAC = 120° and AB =AC = 13 cm. Calculate BC and the area of the triangle.
×
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