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Law of Sines and Law of Sines and CosinesCosines

Trigonometry applied Trigonometry applied tototriangles without right triangles without right angles.angles.

You have learned to apply You have learned to apply trigonometry to right trigonometry to right angled triangles.angled triangles.

A

hyp

adj

opp

adjopptanA

hypadjcosA

hypoppsinA

Now we extend our Now we extend our trigonometry so that we trigonometry so that we can deal with triangles can deal with triangles which are not right which are not right angled.angled.

First we introduce the following notation.First we introduce the following notation. We use capital letters for the angles, We use capital letters for the angles,

and lower case letters for the sides.and lower case letters for the sides.

Q

q

p

r

R

P

A

a

b

c

C

B In ABC The side opposite angle A

is called a. The side opposite angle B

is called b.

In PQR The side opposite angle P

is called p.And so on

There are two new rules.There are two new rules.

1. The Law of 1. The Law of SinesSines

A

ab

C

cB

.

Find the length of BC.Find the length of BC.

Substitute A = 35o, B = 95o, b = 6.2:

Multiply by sin35o:

A

a

6.2 cm

c

C

B

35o

95o

one for finding a side,one for finding a side,one for finding an angle.one for finding an angle.

There are two main There are two main ways of writing the Law ways of writing the Law of Cosinesof Cosines

Law of Cosines

The Law of Cosines (to The Law of Cosines (to find the length of a side)find the length of a side)

A

a

b

c

C

B

The cosine rule for The cosine rule for finding an anglefinding an angle

To use the sine rule you need to know To use the sine rule you need to know an angle and the side opposite it. You an angle and the side opposite it. You can use it to find a side (opposite a can use it to find a side (opposite a second known angle) or an angle second known angle) or an angle (opposite a second known side).(opposite a second known side).

To use the cosine rule you need to To use the cosine rule you need to knowknoweither two sides and the included either two sides and the included angle or all three sides.angle or all three sides.

How do I know whether to use How do I know whether to use the sine rule or the cosine rule?the sine rule or the cosine rule?