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10-1: The Law of Cosines

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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 Ccos-1(.92685) = C

68.89 = 625 – 600 cos C22.05° = 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.

- 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?

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

- Use the law of cosines

1st

x

130

125°

2nd

200

Station

- Assignment
- Page 622
- Problems 1 – 25, odds
- Show work