6.6 Adjusting the Pythagorean Theorem: The Cosine Law
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Transcript of 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
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
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
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
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
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
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… 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