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?. 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

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10 1 the law of cosines

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 cosines1

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:

    • 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


10 1 the law of cosines2

10-1: The Law of Cosines

  • 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.


10 1 the law of cosines3

10-1: The Law of Cosines

  • 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


10 1 the law of cosines4

10-1: The Law of Cosines

  • 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


10 1 the law of cosines5

10-1: The Law of Cosines

  • 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


10 1 the law of cosines6

10-1: The Law of Cosines

  • 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


10 1 the law of cosines7

10-1: The Law of Cosines

  • Assignment

    • Page 622

    • Problems 1 – 25, odds

    • Show work


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