1. The Sine Rule

2. Trigonometry applied to triangles without right angles.
Sine and Cosine rules
Trigonometry applied to
triangles without right angles.

3. Introduction
You have learnt to apply trigonometry to right angled triangles. A
hyp
adj
opp

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

5. First we introduce the following notation.
We use capital letters for the angles, and lower case letters for the sides. A a b c C B
In DABC
The side opposite angle A is called a.
The side opposite angle B is called b. Q q p r R P
In DPQR
The side opposite angle P is called p.
And so on

6. The sine rule Draw the perpendicular from C to meet AB at P.
Using DBPC: PC = a sinB.
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.

7. SOH/CAH/TOA can only be used for right-angled triangles.
The Sine Rule can be used for any triangle: C
The sides are labelled to match their opposite angles b a A B c a
sinA b
sinB c
sinC
= =
The Sine Rule:

8. A
Example 1:
Find the length of BC
76º c
7cm b
63º C x B a
Draw arrows from the sides to the opposite angles to help decide which parts of the sine rule to use. a
sinA c
sinC = x
sin76º 7
sin63º
sin76º × =
× sin76º 7
sin63º
x =
× sin76º
x = cm

9. G 1. B 3. 2. F
53º
13 cm
41º x
8.0
35.3
5.5 x A
62º x
28º
130º D E
5 cm
63º
76º C H
26 mm I 4.
10.7 5.
5.2 cm 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

10. 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

11. P
Example 2:
Find the length of PR
82º x r q
43º
55º Q
15cm R p
Draw arrows from the sides to the opposite angles to help decide which parts of the sine rule to use. p
sinP q
sinQ = 15
sin82º x
sin43º
sin43º × =
× sin43º 15
sin82º
sin43º ×
= x
x = cm

12. 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!
Alternative form of the Sine Rule:
sinA a
sinB b
sinC c
= =

13. C
Example 1:
Find the size of angle ABC
6cm a
4cm b
72º
x º A
Draw arrows from the sides to the opposite angles to help decide which parts of the sine rule to use. B c
sinA a
sinB b =
sin72º 6
sin xº 4
4 × =
× 4
sin72º 6
4 ×
= sin xº
sin xº =
x = sin = º

14. P
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
7 ×
= sin xº
sin xº =
x = sin = º

15. 1. 7.6 cm 2. 3. 47º 82º 105º 6.5cm 5 cm 8.2 cm xº xº xº 8.8 cm 6 cm 5.
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
Slide 10 is incomplete. Try to add slides on applications of Sine Rule 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

16. 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