Sin and Cosine Rules

1
Sin and Cosine Rules

• Objectives calculate missing sides and angles
  is non-right angles triangles

2
Labelling The Triangle
Note Angle A is opposite side a Angle B is
opposite side b Angle C is opposite side c
Vertices (corners) are usually labelled with
capital letters,
Sides are usually labelled with small letters.
3
The Sin Rules
A
c
B
b
a
C
OR Flip it upside down
4
Applying the sin rule
B
Find angle x
a
8 cm
380
1. Make sure your sides are labelled.
C
c
2. Decide whether you are looking for an angle
or side and use the appropriate equation
5 cm
x
b
A
or
To find an angle
3. Identify the information you have and what
part of the equation to use,
5
Applying the formula
B
a
8 cm
380
C
c
5 cm
x
Sin x
Sin 38
b

A
8
5
Sin x 0.123. 8
Sin x 0.123 x 8 0.985
x 80.10
6
Example 2 Using the sin rule
A
Calculate length x
Looking for length
c
9 m
b
x
420
Insert values into equation
B
a
280
C
x sin 42 x 19.17
x 12.83 m to 2 dp.
7
The Cosine Rule
In its most usual form
b2 a2 c2 - 2acCosB
To find a side
A
To find an angle
c
B
b
a
C
8
Rearranging The Formula

• To find any side

b2 a2 c2 - 2acCosB
or
a2 b2 c2 - 2bcCosA
or
c2 a2 b2 - 2abCosC

• To find any angle

or
or
9
Using the formula
Calculate length p
Make sure your triangle is labelled
3.2 cm
p
400
5 cm
Choose the correct equation to use
For side b
For sides
c2 a2 b2 - 2abCosC
b2 a2 c2 - 2acCosB
a2 b2 c2 - 2bcCosA
For angles
10
Substituting into the formula
b2 a2 c2 - 2acCosB
3.2 cm
b2 52 3.22 - 2x5x3.2Cos40
p
400
b2 35.24 - 32Cos40
5 cm
b2 10.73 (2dp)
b 3.3 (1dp)
11
Example 2.

• Calculate angle s

A
7 cm
B
c
b
8 cm
a
12 cm
Cos C 0.828. (3 dp)
s
C 34.10 (1 dp)
C