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{ { Law of Sines and Law of Sines and Cosines Cosines Trigonometry Trigonometry applied to applied to triangles without triangles without right angles. right angles.

{ Law of Sines and Cosines Trigonometry applied to triangles without right angles. 1

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Page 1: { Law of Sines and Cosines Trigonometry applied to triangles without right angles. 1

{{

Law of Sines and Law of Sines and CosinesCosines

Trigonometry applied Trigonometry applied toto

triangles without right triangles without right angles.angles.

Page 2: { Law of Sines and Cosines Trigonometry applied to triangles without right angles. 1

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

A

hyp

adj

opp

adj

opptanA

hyp

adjcosA

hyp

oppsinA

Page 3: { Law of Sines and Cosines Trigonometry applied to triangles without right angles. 1

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.

Page 4: { Law of Sines and Cosines Trigonometry applied to triangles without right angles. 1

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

Page 5: { Law of Sines and Cosines Trigonometry applied to triangles without right angles. 1

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

Page 6: { Law of Sines and Cosines Trigonometry applied to triangles without right angles. 1

1. The Law of 1. The Law of SinesSines

A

ab

C

cB

.

Page 7: { Law of Sines and Cosines Trigonometry applied to triangles without right angles. 1

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

Page 8: { Law of Sines and Cosines Trigonometry applied to triangles without right angles. 1

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

Page 9: { Law of Sines and Cosines Trigonometry applied to triangles without right angles. 1

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

Page 10: { Law of Sines and Cosines Trigonometry applied to triangles without right angles. 1

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

Page 11: { Law of Sines and Cosines Trigonometry applied to triangles without right angles. 1

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 knowknow

either 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?