the sine rule

The Sine Rule

Sine and Cosine rules

Trigonometry applied totriangles without right angles.

2

Introduction

• You have learnt to apply trigonometry to right angled triangles.

3

A

hyp

adj

opp

adj

opptanA

hyp

adjcosA

hyp

oppsinA

• Now we extend our trigonometry so that we can deal with triangles which are not right angled.

4

• First we introduce the following notation.• We use capital letters for the angles,

and lower case letters for the sides.

5

Q

q

p

r

R

P

A

a

b

c

C

B In DABC The side opposite angle

A is called a. The side opposite angle

B is called b.

In DPQR The side opposite angle

P is called p.And so on

The sine ruleDraw the perpendicular

from C to meet AB at P.

6

A

ab

C

cUsing DBPC: PC = a sinB.B

P

Using DAPC: PC = b sinA.

Therefore a sinB = b sinA.Dividing by sinA and sinB gives:

In the same way:

Putting both results together:

The proof needs some changes to deal with obtuse angles.

B

b

A

a

sinsin

C

c

B

b

sinsin

C

c

B

b

A

a

sinsinsin

SOH/CAH/TOA can only be used for right-angled triangles.

The Sine Rule can be used for any triangle:

A B

C

ab

c

The sides are labelled to match their opposite angles

asinA

bsinB

csinC= =The Sine Rule:

Example 1:

C B

A

76º

7cm

Find the length of BC

x

a

sinA

c

sinC

bc

a

=

x

sin76º

7

sin63º= × sin76º

sin76º ×

x =7

sin63º × sin76º

x = 7.6 cm

63º

Draw arrows from the sides to the opposite angles to help decide which parts of the sine rule to use.

Draw arrows from the sides to the opposite angles to help decide which parts of the sine rule to use.

1.

2.

3.

4.

5. 6

.

7.

A

B

C

D E

F

G

H I

P

Q

R

62º

53º

5 cm

x28º

130º

13 cmx

41º

76º

x

26 mm

37º

77º

10 m

x 5.2

cm57º

x62º

x86º

35º 1

2 cm

x

85º

65º

6 km

5.5 8.0

10.7

66º

35.3

63º

61º

5.2

6.9

6.6

Remember:

• Draw a diagram• Label the sides• Set out your working exactly as you have

been shown• Check your answers regularly and ask for

help if you need it

Example 2:

Q R

P

55º

82º

15cm

Find the length of PR

x

p

sinP

q

sinQ

r q

p

=

15

sin82º

x

sin43º= × sin43º

sin43º ×

= x15

sin82ºsin43º ×

x = 10.33 cm

43º

Draw arrows from the sides to the opposite angles to help decide which parts of the sine rule to use.

Draw arrows from the sides to the opposite angles to help decide which parts of the sine rule to use.

Finding an Angle

The Sine Rule can also be used to find an angle, but it is easier to use if the rule is written upside-down!

sinA a

sinBb

sinC c= =

Alternative form of the Sine Rule:

Example 1:

A B

C

72º

6cm

Find the size of angle ABC

x º

sinA

a

sinB

b

ba

c

=

sin72º

6

sin xº

4= × 44 ×

= sin xº4 ×

sin xº = 0.634

Draw arrows from the sides to the opposite angles to help decide which parts of the sine rule to use.

Draw arrows from the sides to the opposite angles to help decide which parts of the sine rule to use.

4cm

sin72º

6

x = sin-1 0.634 = 39.3º

Example 2:

R Q

P

85º

8.2cm

Find the size of angle PRQ

x º

sinP

p

sinR

r

qr

p

=

sin85º

8.2

sin xº

7= × 77 ×

= sin xº7 ×

sin xº = 0.850

7cm

sin85º

8.2

x = sin-1 0.850 = 58.3º

1.2. 3.

4.5.

6. 7.

47º

6 cmxº

5 c

m

xº

105º

8.8 cm

6.5cm

xº

33º

5.2 cm

5.5 cm

xº

7.6 cm

8.2

cm

xº

82º

8 m

70º

9.5

m

(←Be careful!→)

xº

27º

6 km

3.5 km

74º

xº

7 mm

9 mm

37.6°66.6°

45.5°

31.0°

51.1°

57.7°

92.1°

52.3º

22.9º

Remember:

• Draw a diagram• Label the sides• Set out your working exactly as you have

been shown• Check your answers regularly and ask for

help if you need it