Sin and Cosine Rules

Objectives: calculate missing sides and angles is non-right

angles triangles

Labelling The Triangle

B

A

Ca

bc

Vertices (corners) are usually labelled with capital letters,

Sides are usually labelled with small letters.

Note: Angle A is opposite side a

Angle B is opposite side b

Angle C is opposite side c

The Sin Rules

c

SinC

b

SinB

a

SinA

SinC

c

SinB

b

SinA

a

OR

Flip it upside down

A

B

C

c

a

b

Applying the sin rule

5 cm

8 cm380

x

Find angle x

A

B

C

a

b

c

1. Make sure your sides are labelled.

2. Decide whether you are looking for an angle or side and use the appropriate equation

c

SinC

b

SinB

a

SinA

SinC

c

SinB

b

SinA

aorTo find an angle

3. Identify the information you have and what part of the equation to use,

Applying the formula

5 cm

8 cm380

x

A

B

C

a

b

c

c

SinC

b

SinB

a

SinA

Sin x

8

Sin 38

5=

Sin x = 0.123….

8 Sin x = 0.123 x 8 = 0.985

x = 80.10

Example 2: Using the sin rule

28sin

9

42sin

x

28sin

9

42sin

x

A

B

C

a

b

c

420

280

9 m

x

Calculate length x

c

SinC

b

SinB

a

SinA

SinC

c

SinB

b

SinA

a

Looking for length

...17.1942sin

x

Insert values into equation

x = sin 42 x 19.17x = 12.83 m to 2 dp.

The Cosine Rule

b2 = a2 + c2 - 2acCosB

In its most usual form:

To find a side:

To find an angle:

ac

bcaCosB

2

222

A

B

C

c

a

b

Rearranging The Formula

• To find any side:

ab

cbaCosC

2

222

b2 = a2 + c2 - 2acCosB

a2 = b2 + c2 - 2bcCosA

c2 = a2 + b2 - 2abCosC

or

or

ac

bcaCosB

2

222

bc

acbCosA

2

222 or or

• To find any angle:

Using the formula

B

A

C

c

a

b

5 cm

400

3.2 cm

Calculate length p

p

Make sure your triangle is labelled

Choose the correct equation to use:

ab

cbaCosC

2

222

b2 = a2 + c2 - 2acCosB

a2 = b2 + c2 - 2bcCosA c2 = a2 + b2 - 2abCosC

ac

bcaCosB

2

222

bc

acbCosA

2

222

For sides:

For angles:

For side b

Substituting into the formula

B

A

C

5 cm

400

3.2 cmp

c

a

bb2 = a2 + c2 - 2acCosB

b2 = 52 + 3.22 - 2x5x3.2Cos40

b2 = 35.24 - 32Cos40

b2 = 10.73 (2dp)b = 3.3 (1dp)

Example 2.

• Calculate angle s

ab

cbaCosC

2

222

8 cm

12 cm

7 cm

s

AB

C

c

a

b

192

159CosC

8122

7812 222

CosC

Cos C = 0.828…. (3 dp)

C = 34.10 (1 dp)