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The Sine Rule. Sine and Cosine rules. Trigonometry applied to triangles without right angles. hyp. opp. A. adj. Introduction. You have learnt to apply trigonometry to right angled triangles. Now we extend our trigonometry so that we can deal with triangles which are not right angled. B. - PowerPoint PPT Presentation
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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
