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To find unknown sides :
To find unknown angles :
a
b
c


sin A sin B sin C
sin A sin B sin C


a
b
c
SOLUTION OF TRIANGLES
1.2
Use Sine Rule to find the unknown sides or angles of a triangle.
Task 1 : Find the unknown sides of a triangle when two of its angles and one of the corresponding
sides are known.
(1) Diagram 1 shows the triangle ABC.
Answer :
BC
8.2

0
sin 75
sin 400
BC 
Diagram 1
8.2
 sin 750
sin 40 0
Using the scientific calculator,
BC = 12.32 cm
Calculate the length of BC.
(2) Diagram 2 shows the triangle PQR
Diagram 2
[ 8.794 cm ]
Calculate the length of PQ.
(3) Diagram 3 shows the triangle DEF.
D
15 cm
600
E
350 16’
Diagram 3
F
Calculate the length of DE.
[ 10.00 cm ]
(4) Diagram 4 shows the triangle KLM.
L
K
420
0
63
15 cm
Diagram 4
M
Calculate the length of KM.
[ 11.26 cm ]
Solutions of Triangles
1
(5) Diagram 5 shows the triangle ABC.
Answer :
ABC  180 0  40 0  75 0  65 0
AC
8.2

0
sin 65
sin 40 0
BC 
Diagram 5
8.2
 sin 65 0
0
sin 40
Using the scientific calculator,
Calculate the length of AC.
AC = 11.56 cm
(6) Diagram 6 shows the triangle PQR
Diagram 6
Calculate the length of PR.
[ 6.527 cm ]
(7) Diagram 7 shows the triangle DEF.
D
15 cm
600
E
350 16’
Diagram 7
F
Calculate the length of EF.
[ 17.25 cm ]
(8) Diagram 8 shows the triangle KLM.
L
K
420
0
63
15 cm
Diagram 8
M
Calculate the length of KL.
[ 16.26 cm ]
Solutions of Triangles
2
Task 2 : Find the unknown sides of a triangle when two of its angles and the side not corresponding
to the angles are known.
(9) Diagram 9 shows the triangle ABC.
Answer :
ABC  1800  470  780  550
BC
11.2

0
sin 47
sin 550
BC 
11.2
 sin 47 0
0
sin 55
Diagram 9
Using scientific calculator,
Calculate the length of BC.
BC = 9.9996 cm or 10.00 cm
(10) Diagram 10 shows the triangle ABC.
Diagram 10
Calculate the length of AC.
[ 4.517 cm ]
(11) Diagram 11 shows the triangle PQR.
7.2 cm
P
250
0
28
R
Diagram 11
Q
Calculate the length of PQ.
[ 3.810 cm ]
(12) Diagram 12 shows the triangle DEF.
D
720
E
510
5.6 cm
F
Diagram 12
Calculate the length of DE.
[ 5.189 cm ]
Solutions of Triangles
3
Task 3 : Find the unknown angles of a triangle when two sides and a non-included angle are given.
(1) Diagram 1 shows the triangle ABC.
Answer :
A
10 cm
sin C sin 60 0

10
15
15 cm
600
B
sin C 
C
Diagram 1
10 sin 60 0
15
sin C  0.5774
C  sin 1 0.5774
C  35.27 0
Find ACB.
(2) Diagram 2 shows the triangle KLM
15 cm
L
K
9 cm
500
Diagram 2
M
Find KLM
[ 27.360 ]
(3) Diagram 3 shows the triangle DEF.
D
3.5 cm
12.5 cm
430 24’
E
F
Diagram 3
Find DFE.
[ 11.090 ]
(4) Diagram 4 shows the triangle PQR.
13 cm
R
P
10 cm
0
130
Diagram 4
Q
Find QPR.
[ 36.110 ]
Solutions of Triangles
4
(5) Diagram 5 shows the triangle ABC.
Answer :
sin A sin 110 0

9
14
9 sin 1100
sin A 
14
A
14 cm
1100
Diagram 5
B
sin A  0.6041
A  sin 1 0.6041
A  37.160
9 cm
C
Find ABC.
ABC  180 0  110 0  37.16 0
 32.84 0
(6) Diagram 6 shows the triangle KLM.
4.2 cm
K
L
2.8 cm
250
Diagram 6
M
Find KLM.
[ 138.640 ]
(7) Diagram 7 shows the triangle DEF.
E
D
340
4.4 cm
6.7 cm
F
Diagram 7
Find DFE.
[ 124.460 ]
(8) Diagram 8 shows the triangle PQR.
P
12.3 cm
R
550
7.7 cm
Q
Diagram 8
Find PQR.
[ 94.150 ]
Solutions of Triangles
5
Task 4 : Find the unknown side of a triangle when two sides and a non-included angle are given.
(1) Diagram 1 shows the triangle ABC.
Answer :
sin C sin 37 0

14
9
14 sin 37 0
sin C 
9
A
370
14 cm
B
Diagram 1
C
sin C  0.9362
9 cm
C  sin 1 0.9362
Given that ACB is an obtuse angle, find
the length of AC.
C  110.580
B  1800  110.580  370
 32.420
AC
sin 32.42
0

AC 
9
sin 37 0
9 sin 32.42 0
sin 37 0
AC = 8.018 cm
(2) Diagram 2 shows the triangle KLM
9 cm
L
K
0
40
7 cm
M
Diagram 2
Given that KLM is an obtuse angle, find
the length of ML.
[ 2.952 cm ]
Solutions of Triangles
6
(3) Diagram 3 shows the triangle DEF.
D
8 cm
420
E
11 cm
F
Diagram 3
Given that the value of EDF is greater than
900, find the length of DE.
[ 5.040 cm ]
(4) Diagram 4 shows the triangle PQR.
8.5 cm
P
R
460
6.9 cm
Diagram 4
Q
Given that PQR is an angle in the second
quadrant of the cartesian plane, find the
length of QR.
[ 2.707 cm ]
(5) Diagram 5 shows the triangle KLM.
K
L
0
23
17.3 cm
9.2 cm
Diagram 5
M
Given that KLM is an angle in the second
quadrant of the cartesian plane, find the
length of KL.
[ 9.686 cm ]
Solutions of Triangles
7
Solutions of Triangles
8