Sine and Cosine Rules
Objectives: calculate missing
sides and angles is non-right
angles triangles
Labelling The Triangle
Note:
A
Angle A is opposite side a
Angle B is opposite side b
Angle C is opposite side c
c
B
b
C
a
Vertices (corners) are usually labelled with capital letters,
Sides are usually labelled with small letters.
The Sin Rules
A
SinA

a
SinB
b

c
SinC
B
c
b
a
C
OR
Flip it upside
down
a
SinA

b
SinB

c
SinC
Applying the sin rule
B
a
380
Find angle x
8 cm
1. Make sure your
sides are labelled.
C
c
x
2. Decide whether you are
looking for an angle or side
and use the appropriate
equation
5 cm
b
A
To find an angle
SinA
a

SinB
b
3. Identify the information you have and
what part of the equation to use,

SinC
c
or
a
SinA

b
SinB

c
SinC
Applying the formula
B
SinA
a
380

SinB
a
b

SinC
c
8 cm
C
c
x
A
5 cm
b
Sin x
8
=
Sin 38
Sin x = 0.123….
8
Sin x = 0.123 x 8 = 0.985
x = 80.10
5
Example 2: Using the sin rule
A
Calculate length x
c
SinA
a
9m
B
SinB

x
420
x

sin 42
b
9
sin 28
b

Looking for length
a
SinC
SinA
c
Insert
values into
equation
x
sin 42


b

SinB
c
SinC
9
sin 28
a
280
x
C
 19 . 17 ...
sin 42
x = sin 42 x 19.17
x = 12.83 m to 2 dp.
The Cosine Rule
In its most usual form:
To find a side:
b2 = a2 + c2 - 2acCosB
A
To find an angle:
CosB

a
2
 c
2
b
2
c
B
2 ac
b
a
C
Rearranging The Formula
• To find any side:
b2 = a2 + c2 - 2acCosB
or
or
a2 = b2 + c2 - 2bcCosA
c2 = a2 + b2 - 2abCosC
• To find any angle:
CosA

b
2
 c
2
2 bc
 a
2
or
CosB

a
2
 c
2
2 ac
b
2
or
CosC

a
2
 b
2
2 ab
 c
2
Using the formula
Calculate length p
A
c
B
Make sure your triangle is labelled
3.2 cm
b
p
400
5 cm
a
C
Choose the correct equation to use:
For side b
For sides:
For angles:
a2 = b2 + c2 - 2bcCosA
CosA

b
2
b2 = a2 + c2 - 2acCosB
 c
2
2 bc
 a
2
CosB

a
2
 c
2
2 ac
b
c2 = a2 + b2 - 2abCosC
2
CosC

a
2
 b
2
2 ab
 c
2
Substituting into the formula
A
b2 = a2 + c2 - 2acCosB
c
b
3.2 cm
B
p
400
5 cm
b2 = 52 + 3.22 - 2x5x3.2Cos40
b2 = 35.24 - 32Cos40
a
C
b2 = 10.73 (2dp)
b = 3.3 (1dp)
Example 2.
• Calculate angle s
A
7 cm
c
8 cm
CosC
B
CosC
12 cm
CosC
s
C
2
 b
2
 c
2
2 ab
b
a

a

159

12
2
2
8  7
2
2  12  8
Cos C = 0.828…. (3 dp)
192
C = 34.10 (1 dp)