Aim: What is the law of cosine?

Do Now: In ABC, AB = c, BC = a, and AC = b.

y

xA B(c,0)

C(x,y)

ab

c

1. In Δ ACD, find cos A

2. In ΔACD, find sin A

Dx

y

HW: p.555 # 10,16,20,23,24,25

cos Ax

b sin A

y

b

222 )( yxca

x = b cos A y = b sin A

222 )sin()cos( AbAbca AbAbAbcc 22222 sincoscos2

AbccAAb cos2)sin(cos 2222

Abccb cos222

In Δ BCD, we can calculate

a b c bc A2 2 2 2 cos

Law of Cosine

It can also be written as

b a c ac B

c a b ab C

2 2 2

2 2 2

2

2

cos

cos

How and when do we use the law of cosine?

1.In a triangle, if the lengths of any two sides and the included angle measurement are

known, we can apply the law of cosine to find the length of the third side.

2. In a triangle, if the lengths of all three sides are known, we then can apply the law of cosine to find the measure

degree of any angle.

We can use the SAS and SSS to memorize, where letters stand for the known values.

Application:

1. In ∆ABC, if a = 4, c = 6 and cos B = 1/16. Find b.

b2 2 24 6 2 4 6 1 16 16 36 48 1 16 52 3 49 / /

b = 7 (use positive value)

2. In ∆ABC,a = 5, b = 7 and c = 10. Find mB

100

76cos B mB = 40.5°

3. In ∆RST r = 11, s = 12, and mT = 120°,

Find t to the nearest integer

R

ST

t

11

12

120°

t = 20

4. Two airplanes leave an airport at the same time. The first flies 150 km/h in a direction of 320. The second flies 200 km/h in a direction of 200. After 2 hours, how far apart are the planes?

A

B

O

400 km300 km

120°

608 km

A

The beam of a searchlight situated at an offshore point W sweeps back and forth between shore points A and B. Points W is located 12 km from A and 25 km from B. The distance between A and B is 29 km. Find the measure of AWB to the nearest tenth degree.

96.9