The Sine Rule

C. McMinn

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:

We use the Sine Rule when:

We have two angles and a side

OR

We have two sides and the non-included angle (this is the ambiguous case)

Example 1:

C B

A

76º

7cm

Find the length of BC

x

a

sinA

c

sinC

bc

a

=

x

sin76º

7

sin63º= × sin76ºsin76º ×

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.

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= × sin43ºsin43º ×

= 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.

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

77º10 m

x 5.2 cm

57º

x62º

x86º

35º

12 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

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 cm

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

37.666.6

45.5

31.0

51.1

57.7

92.152.3º 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