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What you will learn

How to solve triangles by using the Law of Cosines

How to find the area of triangles if the measures of the three sides are given

Objective: 5-8 The Law of Cosines

2Law of Cosines

When given three side lengths (SSS) or two sides and the included angle (SAS), the law of sines won’t work. For these situations, we have the law of cosines.

a2 = b2 + c2 – 2bc cos A

b2 = a2 + c2 – 2ac cos B

c2 = a2 + b2 – 2ab cos C

Objective: 5-8 The Law of Cosines

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Example for SAS Solve the triangle: A = 120o, b = 9, c = 5Use: a2 = b2 + c2 – 2bc cosA becauseA and not a.

Objective: 5-8 The Law of Cosines

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You Try Solve the triangle: A = 39.4o, b = 12, c =

14

Objective: 5-8 The Law of Cosines

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Another Example SSS Solve the triangle: a = 24, b = 40, c =

18.Angle B is the largest. How do you know?Use b2 = a2 + c2 – 2ac cosB

Objective: 5-8 The Law of Cosines

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You Try Solve the triangle: a = 19, b = 24.3, c =

21.8

Objective: 5-8 The Law of Cosines

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Finding Area – SSS Find the area of triangle ABC if a = 4, b = 7,

and c = 9. Find angle A then use AbcK sin

2

1

Objective: 5-8 The Law of Cosines

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Finding Area – Hero’s Formula If you know three sides of a triangle and no

angle measures, you can use Hero’s Formula to calculate area:

))()(( csbsassK )(2

1cbas

Objective: 5-8 The Law of Cosines

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Example of Hero’s Formula Find the area of triangle ABC with a =

72, b = 83, and c = 95.))()(( csbsassK )(

2

1cbas

Objective: 5-8 The Law of Cosines

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You Try Find the area of ABC with a = 47, b = 53,

and c = 44.

))()(( csbsassK )(2

1cbas

Objective: 5-8 The Law of Cosines

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Homework page 331, 11-23 odd, 25, 28