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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

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