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A
CB
c b
a
The sine rules enables us to calculate sides and angles
In the some triangles where there is not a right angle.
((
The Sine Rule is used to solve any problems involvingtriangles when at least either of the following isknown:
a) two angles and a side
b) two sides and an angle opposite a given side
In Triangle ABC, we use the convention that
a is the side opposite angle A
b is the side opposite angle B
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Example 2 (Given two sides and an included angle)
Solve triangleABCin which A = 55, b = 2.4cm and
c= 2.9cm
By cosine rule,
a2 = 2.42+ 2.92- 2 x 2.9 x 2.4 cos 55
= 6.1858
a=
2.49cm
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EitherC
c
B
b
A
a
sinsinsin!!
Or
c
C
b
B
a
A sinsinsin!!
[1]
[2]
Use [1] when finding a side
Use [2] when finding an angle
Using this label of a triangle,
the sine rule can be stated
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Example:A
CB
c
Given
Angle ABC =600
Angle ACB = 500
Find c.
7cm
To find c use the following proportion:
B
b
C
c
sinsin!
0060sin
7
50sin!
c
0
0
60sin
50sin7 xc !
c= 6.19 ( 3 S.F)
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A
C
B
15 cm
6 cm
1200
BFind
012015
,6
!!
!(
AandcmBC
cmACBACIn
SOLUTION:
a
A
b
B sinsin!
15
120sin
6
sin0
!B
15
60sin6sin
0x
B !
sin B = 0.346
B= 20.30
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SOLVE THE FOLLOWING USING THESINE RULE:
Problem 1 (Given two angles and a side)
In triangleABC, A = 59, B = 39 and a = 6.73cm.
Find angle C, sides b and c.
DRILL:
Problem 2 (Given two sides and an acute angle)
In triangleABC, A = 55, b = 16.3cm and
a = 14.3cm. Find angle B, angle C and side c.
Problem 3 (Given two sides and an obtuse angle)
In triangleABC A =100, b = 5cm and a = 7.7cm
Find the unknown angles and side.
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C= 180 - (39 + 59)= 82
Answer Problem 1
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Bsin
3.16
55sin
3.140!
3.14
55sin3.16sin
0
!B
= 0.9337
00.69!B
0005569180 !C
056!
0056sin69sin
3.16 c!
0
0
69sin
56sin3.16!c
= 14.5 cm (3 SF)
ANSWER PROBLEM 2
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Answer Problem 3
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Sometimes the sine rule is not enough to help us
solve for a non-right angled triangle.
For example:C
B
A
a14
18 300
In the triangle shown, we do not have enough information
to use the sine rule. That is, the sine rule only provided the
Following:
CBa
sin18
sin14
30sin0
!!
Where there are too many unknowns.
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For this reason we derive another useful result, known as the
COSINE RULE. The Cosine Rule maybe used when:
a. Two sides and an included angle are given.
b. Three sides are given
B
C
A
a
b
c
C
B
A
a c
The cosine Rule: To find the length of a sidea2 =b2+ c2 - 2bccosA
b2 =a2 + c2 - 2accosB
c2 =a2 + b2 - 2ab cosC
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THE COSINE RULE:To find an angle when given all three
sides.
bc
acbA
2cos
222
!
ac
bca
B 2cos
222
!
ab
cba
C 2cos
222
!
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Example 1 (Given three sides)
In triangleABC, a = 4cm, b = 5cm andc= 7cm. Find the size of thelargestangle. The largest angle is the onefacing the longest side, which is angleC.
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DRILL:
ANSWER
PAGE203#S 1-10
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