1. The Sine Rule
The Sine Rule is used for cases in which the Cosine Rule cannot be applied. It is used to find:
1. An unknown side, when we are given two angles and a side.
2. An unknown angle when we are given two sides and an angle that is not included.
a2 = b2 + c2 – 2bcCosA 12 18
75o 19
Cosine Rule ? 12 18
50o
Sine Rule
75o
28/08/2018 1

2. The Sine Rule Deriving the rule C b a B A c This can be extended to
Consider a general triangle ABC.
Deriving the rule P
Draw CP perpendicular to BA
This can be extended to
or equivalently
28/08/2018 2

3. To find an unknown side we need 2 angles and a side.
The Sine Rule
To find an unknown side we need 2 angles and a side.
Not to scale a 1.
45o
60o
5.1 cm 2.
63o m
85o
12.7cm
15o 3. p
145o
45 m
28/08/2018 3

4. To find an unknown angle we need 2 sides and an angle not included.
The Sine Rule
To find an unknown angle we need 2 sides and an angle not included.
Not to scale 1.
60o
5.1 cm
4.2 cm x 2.
63o
12.7cm
11.4cm y 3.
145o
45 m
99.7 m z
28/08/2018 4

5. The Sine Rule
Application Problems A D
The angle of elevation of the top of a building measured from point A is 25o. At point D which is 15m closer to the building, the angle of elevation is 35o Calculate the height of the building. T B
35o
25o
10o
36.5
145o
15 m
Angle TDA =
180 – 35 = 145o
Angle DTA =
180 – 170 = 10o
28/08/2018 5

6. The Sine Rule A
The angle of elevation of the top of a column measured from point A, is 20o. The angle of elevation of the top of the statue is 25o. Find the height of the statue when the measurements are taken 50 m from its base B T C
180 – 115 = 65o
Angle BCA =
180 – 110 = 70o
Angle ACT =
180 – 70 = 110o
Angle ATC =
25o
65o
110o
20o
70o
53.2 m 5o
50 m
28/08/2018 6

7. The Sine Rule. (Used for Non-Right Angled Triangles)
Calculating a Length
a = b
Sin A Sin B b
580
7 cm A
The angles are OPPOSITE their sides. B
390 a x
Working Out
Let ‘a’ stand
for the unknown length.
a = a =
9.43 7
Sin 580
Sin 390
Sin 580
Sin 390
a = 7 Sin 580
Sin 390
a = cm 7
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8. Calculate the Missing Lengths. (Answers to 2 d.p.)
x = 7.36 cm 3) x 1)
6 cm
1100
8 m
700
x = 4.06 m
500
300 x 2) 4)
800
400
x= m x
600 x
7 m
x= 4.57 m
500
20 m 8
Answers
Menu 8

9. The Sine Rule. (Used for Non-Right Angled Triangles)
Calculating an Angle.
a = b
Sin A Sin B a b
13.7 cm
6.5 cm B
270
Working Out A x0 =
Sin A Sin 270
13.7 Sin 270 = Sin A
Sin
0.9569
13.7
730 A
6.5 =
6.5
Sin A
Sin 270
= Sin A
Sin = A
730 = A 9
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10. 2) 1) 3) 4) Calculate The Missing Angles. (Answers to 1 d.p.) 35.30
31.70 2) 1)
800
9 cm
7 cm
8 cm x x
480
15 cm 3)
48.90
78.20 4)
700
8.1 cm
9.5 cm
14 cm x
400 x
25 cm 10
Answers
Menu

11. The Sine Rule
The Sine Rule is used for cases in which the Cosine Rule cannot be applied. It is used to find:
1. An unknown side, when we are given two angles and a side.
2. An unknown angle when we are given two sides and an angle that is not included.
a2 = b2 + c2 – 2bcCosA 12 18
75o 19
Cosine Rule ? 12 18
50o
Sine Rule
75o

12. The Sine Rule Deriving the rule C b a B A c This can be extended to
Consider a general triangle ABC.
Deriving the rule P
Draw CP perpendicular to BA
This can be extended to
or equivalently

13. To find an unknown side we need 2 angles and a side.
The Sine Rule
To find an unknown side we need 2 angles and a side.
Not to scale a 1.
45o
60o
5.1 cm 2.
63o m
85o
12.7cm
15o 3. p
145o
45 m

14. To find an unknown angle we need 2 sides and an angle not included.
The Sine Rule
To find an unknown angle we need 2 sides and an angle not included.
Not to scale 1.
60o
5.1 cm
4.2 cm x 2.
63o
12.7cm
11.4cm y 3.
145o
45 m
99.7 m z

15. The Sine Rule
Application Problems A D
The angle of elevation of the top of a building measured from point A is 25o. At point D which is 15m closer to the building, the angle of elevation is 35o Calculate the height of the building. T B
35o
25o
10o
36.5
145o
15 m
Angle TDA =
180 – 35 = 145o
Angle DTA =
180 – 170 = 10o

16. The Sine Rule A
The angle of elevation of the top of a column measured from point A, is 20o. The angle of elevation of the top of the statue is 25o. Find the height of the statue when the measurements are taken 50 m from its base B T C
180 – 110 = 70o
180 – 70 = 110o
Angle ATC =
180 – 115 = 65o
Angle BCA =
Angle ACT =
25o
65o
110o
20o
70o
53.2 m 5o
50 m

17. The Sine Rule
The Sine Rule is used for cases in which the Cosine Rule cannot be applied. It is used to find:
1. An unknown side, when we are given two angles and a side.
2. An unknown angle when we are given two sides and an angle that is not included.
a2 = b2 + c2 – 2bcCosA 12 18
75o 19
Cosine Rule ? 12 18
50o
Sine Rule
75o
28/08/2018 17

18. The Sine Rule Deriving the rule C b a B A c This can be extended to
Consider a general triangle ABC.
Deriving the rule P
Draw CP perpendicular to BA
This can be extended to
or equivalently
28/08/2018 18

19. To find an unknown side we need 2 angles and a side.
The Sine Rule
To find an unknown side we need 2 angles and a side.
Not to scale a 1.
45o
60o
5.1 cm 2.
63o m
85o
12.7cm
15o 3. p
145o
45 m
28/08/2018 19

20. To find an unknown angle we need 2 sides and an angle not included.
The Sine Rule
To find an unknown angle we need 2 sides and an angle not included.
Not to scale 1.
60o
5.1 cm
4.2 cm x 2.
63o
12.7cm
11.4cm y 3.
145o
45 m
99.7 m z
28/08/2018 20

21. The Sine Rule
Application Problems A D
The angle of elevation of the top of a building measured from point A is 25o. At point D which is 15m closer to the building, the angle of elevation is 35o Calculate the height of the building. T B
35o
25o
10o
36.5
145o
15 m
Angle TDA =
180 – 35 = 145o
Angle DTA =
180 – 170 = 10o
28/08/2018 21

22. The Sine Rule A
The angle of elevation of the top of a column measured from point A, is 20o. The angle of elevation of the top of the statue is 25o. Find the height of the statue when the measurements are taken 50 m from its base B T C
180 – 115 = 65o
Angle BCA =
180 – 110 = 70o
Angle ACT =
180 – 70 = 110o
Angle ATC =
25o
65o
110o
20o
70o
53.2 m 5o
50 m
28/08/2018 22

23. 1 1 Trigonometry 5: Sine, cosine and tangent for any angle y
As the Point P moves in an anti-clockwise direction around the circumference of the circle, the angle  changes from 0o to 360o. O 1 P
(x,y) A y 
Consider the right-angled triangle formed by the vertical line PA. x x
In this triangle the distance OA = x.
The distance OP = y. x O 1 P  A y
So point P has co-ordinates (x,y).
(cos ,sin )
sin 
cos 
Therefore x = cos  and y = sin .
So the co-ordinates of P are (cos , sin ).
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24. The Sine Rule
C. McMinn

25. 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:

26. A
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 = cm

27. P
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
sin43º ×
= x
x = cm

28. 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

29. 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

30. 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
= =

31. C
Example 1:
Find the size of angle ABC
6cm a
4cm b
72º
x º 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
4 ×
= sin xº
sin xº =
x = sin = º

32. 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

33. 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

34. 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