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# 10-1: The Law of Cosines - PowerPoint PPT Presentation

10-1: The Law of Cosines. Essential Question: What is the law of cosines, and when do we use it?. 10-1: The Law of Cosines. In any triangle ABC, with side lengths a, b, c – which are opposite their respective angle, the Law of Cosines states: a 2 = b 2 + c 2 – 2bc cos A

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### 10-1: The Law of Cosines

Essential Question: What is the law of cosines,and when do we use it?

• In any triangle ABC, with side lengths a, b, c – which are opposite their respective angle, the Law of Cosines states:

• a2 = b2 + c2 – 2bc cos A

• b2 = a2 + c2 – 2ac cos B

• c2 = a2 + b2 – 2ab cos C

• Basically: It’s just like the Pythagorean Theorem, then subtracting two times those two sides times the cosine of the angle.

• Proof on board

• The law of Cosines can be used to solve triangles in the following cases:

• Given two sides and an angle between them (SAS)

• Given three sides (SSS)

• The law of cosines helps us solve the situations where the law of sines cannot.

• Example 1: Solve a Triangle with SAS Information.

• Solve triangle ABC below

1) Use law of cosines to find c

c2 = a2 + b2 – 2ab cos C

c2 = 162 + 102 – 2(16)(10) cos 110

c2 = 256 + 100 – 320(-0.3420)

c2 = 356 + 109.4464

c2 = 465.4464

c  21.5742

(you can give 21.6 as an answer,

but use 4 digits to continue solving)

C

2) Use law of sines to find A (or B)

110°

10

16

3) Find the last angle

A

B

B = 180 – 110 – 44.2 = 25.8

c

• Example 2: Solve a Triangle with SSS Information

• Solve a triangle where a = 20, b = 15 and c = 8.3

• Use the law of cosines to find any angle

• Use the law of sines to find another angle

• Use common sense to find the third angle

c2 = a2 + b2 – 2ab cos C -556.11 = -600 cos C

8.32 = 202 + 152 – 2(20)(15) cos C .92685 = cos C

68.89 = 400 + 225 – 600 cos C cos-1(.92685) = C

68.89 = 625 – 600 cos C 22.05° = C

C = 180 – 22.1 – 42.7

C = 115.2

• Example 3: The distance between two vehicles

• Two trains leave a station on different tracks. The tracks make an angle of 125° with the station as the vertex. The first train travels at an average speed of 100 km/h, and the second train travels at an average speed of 65 km/h. How far apart are the trains after 2 hours?

• These questions are helped if you draw a diagram.

1st

x

130

125°

2nd

200

Station

• Example 3: The distance between two vehicles

• Use the law of cosines

• x2 = 1302 + 2002 – 2(100)(200) cos 125

• x2 = 16900 + 40000 – 52000 cos 125

• x2 = 86725.975

• x = 294.5

1st

x

130

125°

2nd

200

Station

• Assignment

• Page 622

• Problems 1 – 25, odds

• Show work