A34 TRIG WITH BEARING
QUESTIONS
1
In the diagram, A and B are two points
on a straight coastline.
North
B is due east of A and AB = 7 km.
SEA
The position of a boat at different times
was noted.
A
B
7
LAND
North
(a) At 8 a.m., the boat was at C, where
∧
∧
AC B = 66° and ABC = 48°.
C
66°
Calculate
48°
(i) the bearing of B from C,
A
[1]
B
7
(ii) the distance AC.
[3]
(b) At 9 a.m., the boat was at D, where
∧
AD = 6.3 km and DA B = 41°.
North
D
6.3
Calculate
41°
A
(i) the area of triangle ADB,
7
B
[2]
(ii) the shortest distance from the boat to the coastline.
(c) At 11 a.m., the boat was at E, where
AE = 9 km and BE = 5 km.
[2]
E
North
9
A
Calculate the bearing of E from A.
© UCLES 2006
4024/02/M/J/06
7
5
B
[4]
2
(a)
A
65
C
B
200
From the top of a vertical tower, AB, an observer sees a car at C.
AB = 65 m and CB = 200 m.
Calculate CÂB.
(b)
[2]
N
R
300
750
P
S
The diagram shows three positions at sea, R, P and S.
R is due north of P and S is due east of P.
RP = 300 m and RS = 750 m.
(i) A boat sailed at a constant speed of 5 km/h from R to S.
It was at R at 22 56.
Find the time it reached S.
[3]
(ii) Calculate the bearing of S from R.
[3]
© UCLES 2007
4024/02/O/N/07
3
B
North
A
115°
2.8
L
4.5
42°
H
The diagram shows the positions of a harbour, H, a lighthouse, L, and two buoys A and B.
HAB is a straight line.
The bearing of A from H is 042°.
HA = 4.5 km, AL = 2.8 km and HÂL = 115°.
(a) Find the bearing of
(i) H from A,
[1]
(ii) L from A.
[1]
(b) Calculate
(i) HL,
[4]
(ii) the area of triangle HAL.
[2]
(c) A boat sailed from the harbour along the line HAB.
(i) Calculate the shortest distance between the boat and the lighthouse.
[2]
(ii) The boat sailed at a constant speed of 3 m/s.
Given that the boat reached A at 07 15, find at what time it left the harbour.
© UCLES 2008
[2]
4024/02/M/J/08
4
North
Q
20
P
75°
50°
17
L
The diagram shows two ports, P and Q, and a lighthouse L.
PQ = 20 km, PL = 17 km, QP̂L = 50° and the bearing of Q from P is 075°.
(a) Find the bearing of P from L.
[1]
(b) Calculate QL.
[4]
(c) (i) Calculate PL̂Q.
[3]
(ii) Hence find the bearing of Q from L.
[1]
(d) A boat leaves P and sails in a straight line to Q.
(i) It takes 4 hours and 53 minutes to sail from P to Q.
It arrives at Q at 02 23.
At what time does it leave P?
[1]
(ii) Calculate the shortest distance between the boat and the lighthouse.
© UCLES 2010
[2]
4024/22/M/J/10
5
S and P are two positions at sea and L is a Lighthouse.
S
1200
L
North
1300
P
S is due West of L and P is due South of L.
SL = 1200 m and SP = 1300 m.
(i) Calculate LP.
[2]
(ii) Calculate the bearing of S from P.
[3]
(iii) A boat sailed from S to P.
It left S at 15 56 and reached P at 16 04.
Calculate its speed in kilometres per hour.
© UCLES 2010
[2]
4024/21/O/N/10
North
6
J 125°
15
L
K
The scale drawing shows a map of three towns, J, K and L.
The distance of K from J is 15 km and the bearing of K from J is 125°.
(i) M is due south of J and due west of K.
Calculate the distance, in kilometres, of M from K.
Give your answer correct to 2 decimal places.
Answer .................................. km [2]
(ii) Using measurements from the diagram, find
(a) the bearing of L from J,
Answer ........................................ [1]
(b) the actual distance, in kilometres, of L from J.
Answer .................................. km [1]
© UCLES 2011
4024/21/M/J/11
7
Thediagramshowsthepositions,P,QandR,ofthreebuoys.
ThebearingofQfromPis054o,PQ=250m,QR=340mandPR=160m.
Q
North
250
Its
54°
P
340
160
Bearing 54 110.10
z
164 106
R
(i) CalculatethebearingofRfromP.
Cosa
1602 2502 340
2116071250
cost
277000
275
1
K
cos
2
110.106
Answer ............................................... [4]
(ii) CalculatetheareaoftrianglePQR.
A
21060
sin
18781.165
Answer .......................................... m2[2]
©UCLES2013
4024/21/M/J/13
8
The diagram shows the positions, P, Q, R and S, of four hotels.
S
North
335
3
x tan
x 33822
Q
210°
North
Sfrom P
500
31178
65°
P
a
R
The bearing of Q from P is 065° and the bearing of R from Q is 210°.
t = 90°.
PQ = 500 m, SQ = 335 m and PQS
t .
(a) Calculate PQR
35
Answer
................................................ [1]
Answer
............................................ m [2]
(b) Calculate the shortest distance from P to QR.
Sir 35 Go
h 500Sin35
h 286.788
286 788
(c) Calculate the bearing of S from P.
tank
K
3,3
marks
tan's
2 33.822
Bearing A
S from P
1 Marks
65 33.822
31.178
Answer
© UCLES 2013
................................................ [3]
4024/22/M/J/13
9
(a)
A
31
C
115
B
ABisverticalandCBishorizontal.
AB=31mandCB=115m.
CalculatetheangleofdepressionofCfromA.
Answer ........................................... [3]
(b)
L
J
354
1100
K
JandKaretwopositionsatsea.
ThebaseofalighthouseisatL.
JisdueEastofLandKisdueSouthofL.
KL=354mandKJ=1100m.
t .
(i) Calculate LJK
Answer ........................................... [2]
(ii) HencefindthebearingofKfromJ.
Answer ........................................... [1]
© UCLES 2016
4024/21/M/J/16
10
D
North
27
48° A
55°
19
C
51°
Th
B
The diagram shows the positions of four islands at A, B, C and D.
A is due north of B.
ˆ = 48°, CAB
ˆ = 55° and BĈA = 51°.
DAC
AC = 19 km and AD = 27 km.
(a) Calculate the bearing of D from A.
180 55
283
98
283
Answer .......................................... [1]
(b) Calculate the bearing of A from C.
055
Answer .......................................... [1]
(c) Calculate the distance between A and B.
Sing sing
AB sin74
AB
19
195m51
7,54
15.36
© UCLES 2016
15.36
Answer .................................... km [3]
4024/21/O/N/16
11
(a)
D
15
B
North
12
A
8
C
A, B, C and D are four towns.
B is 12 km due north of A, C is 8 km due east of A and D is 15 km due west of B.
(i) Calculate the distance of B from C.
Answer ..................................... km [2]
(ii) Calculate the bearing of A from D.
Answer .......................................... [3]
(b)
North
North
T
S
The diagram shows the position of a clock tower, T, and a statue, S, drawn to a scale of 1 cm to 75 m.
(i) Using measurements taken from the diagram, find the actual distance between T and S.
Answer ....................................... m [2]
(ii) A fountain, F, is situated 450 m from T on a bearing of 210°.
Draw and label F.
[2]
(iii) Using measurements taken from the diagram, find the bearing of F from S.
Answer ........................................... [1]
© UCLES 2017
4024/21/M/J/17
12
B
North
25
A 220°
38
C
The diagram shows the positions of three towns, A, B and C.
B is due north of A and the bearing of C from A is 220°.
AB = 25 km and AC = 38 km.
(a) Find the bearing of A from C.
Answer ........................................... [1]
(b) Show that BC = 59.4 km correct to 3 significant figures.
[3]
(c) Calculate the bearing of C from B.
Answer ........................................... [4]
© UCLES 2017
4024/21/O/N/17
13 A boat leaves A and travels 12 km to B.
(a) The boat leaves A at 10 25 and travels at an average speed of 15 km/h.
At what time does the boat arrive at B?
Answer .......................................... [2]
(b)
B
North
2
C
12
56°
A
The bearing of B from A is 056°.
B is 2 km due west of C.
Calculate AC.
Answer .................................... km [4]
(c)
D
cliff
B
2 km
C
C is the base of a cliff.
The top of the cliff, D, is vertically above C.
DC is perpendicular to BC and DC = 105 m.
Calculate the angle of elevation of D from B.
Answer .......................................... [2]
© UCLES 2018
4024/21/M/J/18
14
North
A 135°
2.8
C
3.7
42°
B
A yacht sails the triangular route shown.
The bearing of B from A is 135°.
t = 42°.
BC = 3.7 km, AC = 2.8 km and ABC
t = 62.2°, correct to 1 decimal place.
(a) Show that CAB
[3]
(b) Find the bearing of A from C.
Answer ........................................... [2]
(c) The yacht sails from A to B to C to A.
Calculate the total length of the route.
Answer ..................................... km [4]
© UCLES 2018
4024/22/M/J/18
15
North
B
14
NOT TO
SCALE
62°
13
L
8
A
The diagram shows the positions of two ports, A and B, and a lighthouse L.
The bearing of B from L is 062°.
AB = 13 km, BL = 14 km and AL = 8 km.
(a) Calculate the bearing of A from L.
.................................................... [4]
(b) A boat is located at C.
t = 90° .
C is 11 km from B and BCA
The boat travels to port A in a straight line.
Find the distance the boat travels.
.............................................. km [2]
(c) The boat then travels in a straight line from port A to port B.
It travels at an average speed of 3.75 km/h.
Calculate the time taken for the boat to travel from port A to port B.
Give your answer in hours and minutes.
.............. hours .............. minutes [2]
© UCLES 2019
4024/21/M/J/19
16
North
North
B
A
The diagram shows the positions of two boats, A and B, drawn to a scale of 1 : m.
The actual distance between the two boats is 4 km.
(a) Find m, giving your answer correct to 1 significant figure.
m = ................................................... [2]
(b) Measure the bearing of A from B.
.................................................... [1]
(c) A third boat is positioned at C.
C is on a bearing of 120° from A and on a bearing of 195° from B.
Find and label C on the diagram.
[2]
(d) Find, by measurement, the actual distance in kilometres from A to C.
.............................................. km [2]
(e)
H
NOT TO
SCALE
6
70°
B
4
A
The diagram shows the positions of the boats, A and B, and a harbour, H.
t = 70° .
AB = 4 km, AH = 6 km and ABH
(i)
t .
Calculate AHB
t = ................................................... [3]
AHB
(ii)
The boat at A travels in a straight line to the harbour at H.
The average speed of the boat is p km/h.
It takes 12 minutes 20 seconds for the boat to travel from A to H.
Calculate p.
p = ................................................... [3]
© UCLES 2019
4024/22/M/J/19
17
North
A
125°
950
NOT TO
SCALE
520
C
680
B
The diagram shows the positions of three farms A, B and C on horizontal ground.
Farm B is on a bearing of 125° from farm A.
AB = 950 m, BC = 680 m and AC = 520 m.
t = 44.0° , correct to 1 decimal place.
(a) Show that BAC
[3]
(b) Calculate the bearing of A from C.
.................................................... [1]
(c) Farm A and farm B are joined by a straight track AB.
Amira walks along the track from A at a constant speed of 4.6 km/h.
Calculate the time it takes for Amira to walk from A to the point that is closest to farm C.
Give your answer in minutes and seconds, correct to the nearest second.
........... minutes ............ seconds [4]
(d) A helicopter hovers vertically above farm B.
The angle of elevation of the helicopter from farm A is 10.7°.
Calculate the angle of elevation of the helicopter from farm C.
.................................................... [4]
© UCLES 2019
4024/21/O/N/19
18
S
108°
146
NOT TO
SCALE
North
R
325
42°
P
38°
280
Q
A field is in the shape of a quadrilateral PQRS.
A path crosses the field from P to R.
PQ = 280 m, RS = 146 m and PR = 325 m.
t = 108° and RPQ
t = 38° .
S is on a bearing of 042° from P, PSR
(a) Calculate the bearing of R from P.
.................................................... [4]
(b) (i)
Show that QR = 202 m, correct to the nearest metre.
[3]
(ii)
Mia walks at a constant speed of 5.5 km/h.
Calculate the time it takes her to walk from Q to R.
Give your answer in minutes and seconds, correct to the nearest second.
............ minutes ............ seconds [3]
© UCLES 2019
4024/22/O/N/19