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IGCSE / Math Worksheet / Trigonometry / D.J 1
IGCSE Mathematics Worksheet: trigonometry
1.
pavement
entrance
5°
3.17 mh m
NOT TO SCALE
A shop has a wheelchair ramp to its entrance from the pavement.
The ramp is 3.17 metres long and is inclined at 5° to the horizontal.
Calculate the height, h metres, of the entrance above the pavement.
Show all your working.
Answer ……….………………….…… m [2]
2. A square ABCD, of side 8 cm, has another square, PQRS, drawn inside it.
P, Q, R and S are at the midpoints of each side of the square ABCD, as shown in the diagram.
A B
D C
P
S
R
Q
NOT TO SCALE
(a) Calculate the length of PQ.
Answer (a) ……….………………….…… cm [2]
(b) Calculate the area of the square PQRS.
Answer (b) ……….………………….…… cm2
[1]
IGCSE / Math Worksheet / Trigonometry / D.J 2
3. A plane flies from Auckland (A) to Gisborne (G) on a bearing of 115°.
The plane then flies on to Wellington (W). Angle AGW = 63°.
115°
63°
A
G
North
North
W
400 km
410 km
NOT TO SCALE
(a) Calculate the bearing of Wellington from Gisborne.
Answer (a) ………..………………….…… [2]
(b) The distance from Wellington to Gisborne is 400 kilometres.
The distance from Auckland to Wellington is 410 kilometres.
Calculate the bearing of Wellington from Auckland.
Answer (b) ………..………………….…… [4]
4.
A B
D C
O
E
NOT TO SCALE
A, B, C and D lie on a circle, centre O, radius 8 cm.
IGCSE / Math Worksheet / Trigonometry / D.J 3
AB and CD are tangents to a circle, centre O, radius 4 cm.
ABCD is a rectangle.
(a) Calculate the distance AE.
Answer (a) AE = …………….………… cm [2]
(b) Calculate the shaded area.
Answer (b) ………………….………… cm2
[3]
5. In the diagram below ABD is a straight line.
AB = 4 m and AC = 6 m. Angle BAC = 90°.
A DB
C
4 m
6 m
NOT TO SCALE
(a) (i) Use trigonometry to calculate angle ABC.
Answer (a)(i) Angle ABC = ……….……… [2]
(ii) Find angle CBD.
Answer (a)(ii) Angle CBD = ……………… [1]
(b) Calculate the length of BC.
Answer (b) BC = …………….…….…… m [2]
(c) Work out the perimeter and area of triangle ABC.
Give the correct units for each.
Answer (c) Perimeter = ………….… Area = ………….…[3]
IGCSE / Math Worksheet / Trigonometry / D.J 4
6.
30°
55°
North
P R
Q
S
7 km
15 km
14 km
NOT TO SCALE
The quadrilateral PQRS shows the boundary of a forest.
A straight 15 kilometre road goes due East from P to R.
(a) The bearing of S from P is 030° and PS = 7 km.
(i) Write down the size of angle SPR. [1]
(ii) Calculate the length of RS. [4]
(b) Angle RPQ = 55° and QR = 14 km.
(i) Write down the bearing of Q from P. [1]
(ii) Calculate the acute angle PQR. [3]
(iii) Calculate the length of PQ. [3]
(c) Calculate the area of the forest, correct to the nearest square kilometre. [4]
IGCSE / Math Worksheet / Trigonometry / D.J 5
7. Bashira lives in town A and works in town B, which is 13 kilometres from A on a bearing of
040°. She drives from home to work and then drives to visit her mother who lives in town C.
Town C is 17 kilometres from B on a bearing of 130° from B.
40°
q°p°
130°
13 km
17 km
A
B
C
North
North
North
NOT TO SCALE
(a) By writing down the values of p and q, show that angle ABC = 90°.
Answer (a) p = ………… and q = ……… [1]
(b) Use trigonometry to calculate the size of angle ACB.
Answer (b) Angle ACB = ……..………… [2]
(c) Calculate the distance CA.
Answer (c) CA = ……....………………… [2]
(d) Calculate the area of the triangle ABC.
Answer (d) …..………...…………… km2
[2]
(e) Work out the bearing of A from C.
Answer (e) …….……...………………… [2]
IGCSE / Math Worksheet / Trigonometry / D.J 6
A
B
North
140°
50 km
8.
1400 km
1600 km
13°
36°
95°
L
H
J
W
North
NOT TO SCALE
The diagram shows the positions of four cities in Africa, Windhoek (W), Johannesburg (J),
Harari (H) and Lusaka (L).
WL = 1400 km and WH = 1600 km.
Angle LWH = 13°, angle HWJ = 36° and angle WJH = 95°.
(a) Calculate the distance LH. [4]
(b) Calculate the distance WJ. [4]
(c) Calculate the area of quadrilateral WJHL. [3]
(d) The bearing of Lusaka from Windhoek is 060°.
Calculate the bearing of
(i) Harari from Windhoek, [1]
(ii) Windhoek from Johannesburg. [1]
(e) On a map the distance between Windhoek and Harari is 8 cm.
Calculate the scale of the map in the form 1 : n. [2]
9.
NOT TO SCALE
IGCSE / Math Worksheet / Trigonometry / D.J 7
A ship travels 50 kilometres from A to B on a bearing of 140°, as shown in the diagram.
Calculate how far south B is from A.
Answer ………..…………… km [3]
10.
p°
32 m
22 m
NOT TO SCALE
The height of a tree is 22 metres.
The shadow of the tree has a length of 32 metres.
Calculate the value of the angle marked p° in the diagram.
Answer p = ……….…………… [2]
11.
A x cm
3x cm
26 cm
B
C
120
NOT TO SCALE
In triangle ABC, AB = 3x cm, AC = x cm, BC = 26 cm and angle BAC = 120°.
Calculate the value of x.
Answer x = ……….….………… [3]
IGCSE / Math Worksheet / Trigonometry / D.J 8
12.
98°
QR
S
P13.5 km
10.3 km
7.2 km
North
NOT TO SCALE
P, Q, R and S are ferry ports on a wide river, as shown in the diagram above.
A ferry sails from P, stopping at Q, R and S before returning to P.
(a) Q is 7.2 kilometres due south of P and R is 10.3 kilometres due east of Q.
(i) Show by calculation that angle QPR = 55°.
Answer (a)(i) ……..……………[2]
(ii) Write down the bearing of R from P.
Answer (a)(ii) …….……………[1]
(b) The bearing of S from P is 098° and SP = 13.5 km.
(i) Explain why angle RPS = 27°.
Answer (b)(i) …….……..………[1]
(ii) Angle PRS = 90°. Calculate the distance RS.
Answer (b)(ii) RS = ………… km[2]
(iii) Find the total distance the ferry sails.
Answer (b)(iii) ……………… km[1]
(c) The total sailing time for the ferry is 4 hours 30 minutes.
Calculate the average sailing speed, in kilometres per hour, for the whole journey.
Answer (c) ………………… km/h [2]
IGCSE / Math Worksheet / Trigonometry / D.J 9
13.
O
AB
C
D
E F Q
G
H
P12 cm NOT TO SCALE
A circle, centre O, touches all the sides of the regular octagon ABCDEFGH shaded in the
diagram.
The sides of the octagon are of length 12 cm.
BA and GH are extended to meet at P. HG and EF are extended to meet at Q.
(a) (i) Show that angle BAH is 135°. [2]
(ii) Show that angle APH is 90°. [1]
(b) Calculate
(i) the length of PH, [2]
(ii) the length of PQ, [2]
(iii) the area of triangle APH, [2]
(iv) the area of the octagon. [3]
(c) Calculate
(i) the radius of the circle, [2]
(ii) the area of the circle as a percentage of the area of the octagon. [3]
IGCSE / Math Worksheet / Trigonometry / D.J 10
14. sin x° = 0.707107 and 0 ≤ x ≤ 180.
Find the two values of x.
Answer x = …… or x =………… [2]
15.
E D
F
C
B
18°
25°
12 m
55 m
A
NOT TO SCALE
ABCD represents a building with a vertical flagpole, AF, on the roof.
The points E, D and C are on level ground. EA = 55 metres.
The angle of elevation of A from E is 18° and the angle of elevation of F from E is 25°.
(a) Calculate
(i) ED,
Answer (a)(i) ……..………… m[2]
(ii) FD,
Answer (a)(ii) ……..………… m[2]
(iii) DA.
Answer (a)(iii) ……….…………[2]
(b) Show that AF = 7.4 metres, correct to 1 decimal place.
Answer(b) ……..….….…………[1]
(c) The width, AB, of the building is 12 metres.
The top of the flagpole is attached to the point B by a rope.
Calculate
(i) the length of the rope, FB,
Answer (c)(i) ……..…….…… m[2]
(ii) the angle of elevation of F from B.
Answer (c)(ii) ……..……………[2]
IGCSE / Math Worksheet / Trigonometry / D.J 11
16.
North
North
A
B
C
80° 115°40 km
60 km
Island
NOT TO SCALE
To avoid an island, a ship travels 40 kilometres from A to B and then 60 kilometres from B to C.
The bearing of B from A is 080° and angle ABC is 115°.
(a) The ship leaves A at 11 55.
It travels at an average speed of 35 km/h.
Calculate, to the nearest minute, the time it arrives at C. [3]
(b) Find the bearing of
(i) A from B, [1]
(ii) C from B. [1]
(c) Calculate the straight line distance AC. [4]
(d) Calculate angle BAC. [3]
(e) Calculate how far C is east of A. [3]
17.
NOT TO SCALE
Mahmoud is working out the height, h metres, of a tower BT which stands on level ground.
IGCSE / Math Worksheet / Trigonometry / D.J 12
He measures the angle TAB as 25°.
He cannot measure the distance AB and so he walks 80 m from A to C, where angle ACB = 18°
and angle ABC = 90°.
Calculate
(a) the distance AB,
Answer (a) ……………….……… m[2]
(b) the height of the tower, BT.
Answer (b) ……….……………… m
18.
North
North
North
23°
126°
B
A
P
250 m
NOT TO SCALE
The diagram shows three straight horizontal roads in a town, connecting points P, A and B.
PB = 250 m, angle APB = 23° and angle BAP = 126°.
(a) Calculate the length of the road AB.
Answer (a) AB = ………………. m [3]
(b) The bearing of A from P is 303°.
Find the bearing of
(i) B from P,
Answer (b)(i) ……………………… [1]
(ii) A from B.
Answer (b)(ii) ……………………… [2]
