The Cosine Rule

The Cosine Rule

Can be used with ANY

Can be used with ANY

triangle, NOT just with right

triangle, NOT just with right

triangles!!! triangles!!!

What do you

What do you have to know to

have to know to use the Cosine

use the Cosine Rule?Rule?

Two sides and the included angle (SAS)

C

A

B

c

a

b

B

A

C

What do you

What do you have to know to

have to know to use the Cosine

use the Cosine Rule?Rule?

OR all three sides (SSS)

CB

c

a

b

B

A

C

The C

osi

ne R

ule

The C

osi

ne R

ule

is

:is

:CB

c

a

b

B

A

C

IN O

TH

ER

WO

RD

S:

IN O

TH

ER

WO

RD

S:

(One si

de)2

=

(other side)2 +(la

st sid

e)2

-

2(other side)(la

st sid

e)cos(a

ngle of 1st

side)

Find the remaining sides

and angles in the triangle

using the Cosine Rule

What are we given?

2 sides and the included angle – SAS:

c, b, A

Label sides and angles

a

b

c B

CA

Find the remaining sides

and angles in the triangle

using the Cosine Rule

Let’s find side ‘a’ first a

b

c B

CA

Which version of the Cosine Rule do we want to use?

( )2=( )2+( )2-2( )( )cos( )

a b c b c A

Find the remaining sides

and angles in the triangle

using the Cosine Rule

a

b

c B

CA

Plug in and Solve:( )2=( )2+( )2-2( )( )cos(

)a 1

311 13 1

142

a=8.80cm

8.80 cm

Find the remaining sides

and angles in the triangle

using the Cosine Rule

a

b

c B

CA

What do we have now?

3 sides – SSS: a,b,c

What do we still need to find?

B and C

Find the remaining sides

and angles in the triangle

using the Cosine Rule

Let’s find side B next a

b

c B

CA

Which version of the Cosine Rule do we want to use?

( )2=( )2+( )2-2( )( )cos( )

b a c a c B

8.80 cm

Find the remaining sides

and angles in the triangle

using the Cosine Rule

a

b

c B

CA

Plug in and Solve:( )2=( )2+( )2-2( )( )cos(

)8.8 11 8.8 1

1B

B=81.3

8.80 cm 1

3

Find the remaining sides

and angles in the triangle

using the Cosine Rule

Finally let’s find side C

a

b

c B

CA

Which version of the Cosine Rule do we want to use?

( )2=( )2+( )2-2( )( )cos( )

c a b a b C

8.80 cm

Find the remaining sides

and angles in the triangle

using the Cosine Rule

a

b

c B

CA

Plug in and Solve:( )2=( )2+( )2-2( )( )cos(

)8.8 13 8.8 1

3C

C=56.8

8.80 cm 1

1

YOU DO:YOU DO:Find the remaining sides and

Find the remaining sides and

angles of the triangle using the

angles of the triangle using the Cosine Rule.

Cosine Rule.