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### The Sine Rule

C. McMinn

sinA

b

sinB

c

sinC

= =

SOH/CAH/TOA can only be used for right-angled triangles.The Sine Rule can be used for any triangle:

C

b

The sides are labelled to match their opposite angles

a

A

B

c

The Sine Rule:

Example 1:

Find the length of BC

76º

c

7cm

b

63º

C

x

B

a

a

sinA

c

sinC

=

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

x

sin76º

7

sin63º

sin76º ×

=

× sin76º

7

sin63º

x =

×sin76º

x = 7.6 cm

Example 2:

Find the length of PR

82º

x

r

q

43º

55º

Q

15cm

R

p

p

sinP

q

sinQ

=

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

15

sin82º

x

sin43º

sin43º ×

=

× sin43º

15

sin82º

= x

sin43º ×

x = 10.33 cm

B

3.

1.

2.

F

53º

13 cm

41º

x

8.0

35.3

5.5

x

62º

A

x

130º

28º

D

E

5 cm

63º

76º

C

H

26 mm

I

10.7

4.

5.2 cm

5.

x

61º

R

6.

P

37º

66º

57º

10 m

35º

x

5.2

77º

62º

Q

12 cm

6 km

85º

7.

x

6.6

65º

86º

x

6.9

- Remember:
- Draw a diagram
- Label the sides
- Set out your working exactly as you have been shown
- Check your answers regularlyand ask for help if you need it

sinA

a

sinB

b

sinC

c

= =

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

Alternative form of the Sine Rule:

Example 1:

Find the size of angle ABC

6cm

a

4cm

b

x º

72º

A

B

c

sinA

a

sinB

b

=

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

sin72º

6

sin xº

4

4×

=

× 4

sin72º

6

= sin xº

4×

sin xº = 0.634

x = sin-1 0.634 = 39.3º

Example 2:

Find the size of angle PRQ

85º

q

7cm

r

x º

R

p

8.2cm

Q

sinP

p

sinR

r

=

sin85º

8.2

sin xº

7

7×

=

× 7

sin85º

8.2

= sin xº

7×

sin xº = 0.850

x = sin-1 0.850 = 58.3º

1.

2.

3.

82º

105º

6.5cm

47º

5 cm

8.2 cm

66.6°

xº

37.6°

xº

45.5°

xº

8.8 cm

6 cm

5.

6 km

4.

5.5 cm

31.0°

xº

27º

3.5 km

51.1°

xº

5.2 cm

33º

7.

6.

8 m

74º

57.7°

xº

70º

9 mm

9.5 m

92.1°

xº

52.3º

(←Be careful!→)

22.9º

7 mm

- Remember:
- Draw a diagram
- Label the sides
- Set out your working exactly as you have been shown
- Check your answers regularlyand ask for help if you need it

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