6.6 Adjusting the Pythagorean Theorem: The Cosine Law

Chapter 6 Investigating Non-Right Triangles as Models for Problems: 6.6 Adjusting the

Pythagorean Theorem: The Cosine Law

6.6 Adjusting the Pythagorean Theorem: The Cosine Law

Goal for Today:• Learn about and apply the cosine law

6.6 Adjusting the Pythagorean Theorem: The Cosine Law

Bacca

Abccb

Cabba

cos2b

cos2a

cos2c

*ABC any For

222

222

222

*The same holds true for any triangle Ex. XYZ

6.6 Adjusting the Pythagorean Theorem: The Cosine Law

• The cosine law is used to find the 3rd side of a triangle when 2 sides and a contained angle are known, or

• To find an angle when the length of 3 sides are known

6.6 Adjusting the Pythagorean Theorem: The Cosine Law

• 2 sides and a contained angle… ex. 1A

CB

7cm

5cm

43⁰

?

6.6 Adjusting the Pythagorean Theorem: The Cosine Law

6.6 Adjusting the Pythagorean Theorem: The Cosine Law

A

CB

7cm

5cm

43⁰

?

8.4

8.22

2.5174

)7314.0(7074

43cos704925

43cos)7)(5(275

cos2c

2

2

2

2

222

222

c

c

c

c

c

c

Cabba

Humour Break

6.6 Adjusting the Pythagorean Theorem: The Cosine Law

• 3 sides and finding an angle… ex. 2A

CB

7cm

5cm

4.8

?

6.6 Adjusting the Pythagorean Theorem: The Cosine Law

• 3 sides and finding an angle… ex. 2

?

6.6 Adjusting the Pythagorean Theorem: The Cosine Law

2.43

cos728.070

cos70

70

96.50

cos707404.23

cos70492504.23

cos)7)(5(2758.4

cos2c222

222

C

C

C

C

C

C

Cabba

A

CB

7cm

5cm

4.8

Homework

• Tuesday, January10th - Hwk 6.6 Cosine Law Hwk p.566, #2-10, 12a, 13ac

6.6 Adjusting the Pythagorean Theorem: The Cosine Law

• Ex. 1 A bicycle race follows a triangular course. The three legs of the race are, in order, 2.3km, 5.9km, and 6.2km. Find the angle between the starting leg and the finishing leg to the nearest degree.

6.6 Adjusting the Pythagorean Theorem: The Cosine Law

Ex. 1

?

R

6.2km

Q

P

5.9km

2.3km

6.6 Adjusting the Pythagorean Theorem: The Cosine Law

Ex. 1

6.6 Adjusting the Pythagorean Theorem: The Cosine Law

8.71

3128.0cos

cos52.28

52.28

52.28

92.8

cos52.2892.8

cos52.2873.4381.34

cos52.2873.4381.34

cos52.2829.544.3881.34

cos)3.2)(2.6(2)3.2()2.6()9.5(

cos2p222

222

P

P

P

P

P

P

P

P

Pqrrq

Humour Break

6.6 Adjusting the Pythagorean Theorem: The Cosine Law

• Ex. 2 The radar screen of an airport control tower shows that two plans are at the same altitude. According to the range finder, one plane is 100 km away, in the direction N60°E. The other is 160km away, at a direction of S50°E. How far apart are the two planes?

6.6 Adjusting the Pythagorean Theorem: The Cosine Law

Ex. 2 N

S

N60°E

S50°E

50°

60°100km

160km

C

B

A

6.6 Adjusting the Pythagorean Theorem: The Cosine Law

Ex. 2

6.6 Adjusting the Pythagorean Theorem: The Cosine Law

• Ex. 2… In order to find how far apart the two planes are, we first have to find out the angle opposite the side of the line between the two planes that will be the third side of the triangle…

• We can use the supplementary angle rule…• Angle BCA = 180°- 60°- 50°= 70°

6.6 Adjusting the Pythagorean Theorem: The Cosine Law

km 157

24656

1094435600

)3420.0)(160)(100(22560010000

70cos)160)(100(2160100

cos2c

2

2

2

222

222

c

c

c

c

c

Cabba

Homework

• Thursday, January 9th -

6.6 Cosine law applications Hwk p.568, #14-21, 24 & 26