The Law of SinesThe Law of Cosines

And we’re not talking traffic (7.1, 7.2)

Review how we got the Law of SinesDraw a large triangle, and label vertices A, B, and C.

Be neat– make the sides as straight as you can.

Again neatly, sketch an altitude from vertex B, and label this altitude h.

What relationship is there between h and angle C? (You may want to consider a trig ratio.)

What is the area of a triangle? How could you write it with a trig ratio?

POD– a hands-on experience.

Area = ½ (base)(height).

In this triangle, area would be ½ ba(sinC).

But wait! Draw a height from vertex C. What happens then?

Draw a height from vertex A. What about that?

POD– a hands-on experience.

No matter how we orient the triangle, the area will always be ½ (base)(height). So

BacAbcCabarea sin2

1sin

2

1sin

2

1

Moving on

Start with this.

Multiply each term by 2 and divide each term by abc.

Simplify, what is the final result?

BacAbcCab sin2

1sin

2

1sin

2

1

abc

Bac

abc

Abc

abc

Cab sinsinsin

The Law of Sines

You’ve just built the Law of Sines.

or

What does this tell us?

It is true for all angles, not just acute ones.

b

B

a

A

c

C sinsinsin

B

b

A

a

C

c

sinsinsin

The Law of Sines

You’ve just built the Law of Sines.

What does this tell us?

The ratio between the length of a side in a triangle, and the sine of the opposite angle is constant in that triangle.

Is this cool or what?

b

B

a

A

c

C sinsinsin

Use it

You can use the Law of Sines to solve triangles when given AAS, ASA, or SSA. (What does that mean?) (What is the caution?)

Solve ΔABC given α = 48°, γ = 57°, and b = 47. (What condition is this?)

Draw a diagram if it helps.

Use it

You can use the Law of Sines to solve triangles when given AAS, ASA, or SSA.

Solve ΔABC given α = 48°, γ = 57°, and b = 47.

ASA: two angles and the side between

The third angle is a snap. Then use Law of Sines.

Use it

You can use the Law of Sines to solve triangles when given AAS, ASA, or SSA.

Solve ΔABC given α = 48°, γ = 57°, and b = 47.

β = 180° - 48° - 57°

75sin

47

57sin

75sin

47

48sin

c

a

Use it

You can use the Law of Sines to solve triangles when given AAS, ASA, or SSA.

Solve ΔABC given α = 48°, γ = 57°, and b = 47. (What condition is this?)

β = 75° a = 36 c = 41

Use it

We’ve studied bearing and we’re closing in on vectors. Read p. 535, example 5.

What is the total distance run?

Use it

Read p. 535, example 5. What is the total distance run?

Draw a diagram. 7045

253.0km

Q

P

R

Use it

Read p. 535, example 5. What is the total distance run?

7045

253.0km

Q

P

R

kmq

q

8.145sin

25sin0.3

45sin

0.3

25sin

kmp

p

0.445sin

110sin0.3

45sin

0.3

110sin

Use it

Read p. 535, example 5. What is the total distance run?

The total distance is

1.8 + 4.0 = 5.8 km.

7045

253.0km

Q

P

R

Law of Cosines

We have three ways to write it. Here are two. What is the third?

Baccab

Abccba

cos2

cos2

222

222

New

We have three ways to write it.

What is the pattern?

What triangles would we use this tool for?

What happens if the angle is 90°?

Cabbac

Baccab

Abccba

cos2

cos2

cos2

222

222

222

New

We have three ways to write it.

What is the pattern?

What triangles would we use this tool for? SSS, SAS, SSA

What happens if the angle is 90°?

Cabbac

Baccab

Abccba

cos2

cos2

cos2

222

222

222

Use it

Work with an SSS condition.

If ΔABC has sides a = 90, b = 70, and c = 40, find the three angles.

Use it

Work with an SSS condition.

If ΔABC has sides a = 90, b = 70, and c = 40, find the three angles.

Start with the smallest angle (opposite which side?) to make sure to deal with an acute angle— no ambiguity if you use the Law of Sines later.

Use it

Work with an SSS condition.

If ΔABC has sides a = 90, b = 70, and c = 40, find the three angles.

Next, find the middle angle, since it has to be acute as well.

2.25

9048.cos

cos1260011400

cos12600490081001600

cos70902709040 222

C

C

C

C

C

Use it

Work with an SSS condition.

If ΔABC has sides a = 90, b = 70, and c = 40, find the three angles. You could also use the Law of Sines.

2.48

6667.cos

cos72004800

cos7200160081004900

cos40902409070 222

B

B

B

B

B

Use it

Work with an SSS condition.

If ΔABC has sides a = 90, b = 70, and c = 40, find the three angles.

How could you check your answer?

6.1062.482.25180

2.25

2.48

A

C

B

The Proof

Let’s start by looking at an obtuse triangle in standard position. What are h and k?

c

a

bh

C (k, h)

A B (c, 0)K (k, 0)

The ProofLet’s start by looking at an obtuse triangle in standard

position. k = bcosα h = bsinα

(Why do we multiply by b?)

c

a

bh

C (k, h)

A B (c, 0)K (k, 0)

The Proof

Now for the algebra. We’ll look at right ΔKBC.

c

a

bh

C (k, h)

A B (c, 0)K (k, 0)

cos2

cos2)cos(sin

coscos2sin

)cos()cos(2)sin(

2)(

sin

cos

222

22222

222222

2222

222222

bccba

bccba

bbccba

bbccba

kckchkcha

bh

bk