8-6The Law of Cosines

ObjectiveTo apply the Law of Cosines

Essential UnderstandingIf you know the measures of two side lengths

and the measure of the included angle (SAS), or all three side lengths (SSS), then you can find all the other measures of the

triangle.

A farmer needs to put a pipe through a hill for irrigation. The farmer attaches a 14.5 meter rope and an 11.2 meter rope at each entry point of the pipe and makes a triangle. The ends meet at a 580 angle.

What is the length of the pipe the farmer needs?

14.5 m11.2 m

580

Can you use Law of Sines?

No, you don’t know the angles that are opposite the sides

Pythagorean Theorem?

Not a right triangle

x

Law of Cosines

For any triangle ABC, the Law of Cosines relates the cosine of each angle to the side lengths of the triangle.

c2 = a2 + b2 − 2abcosC

b2 = a2 + c2 − 2accosB

a2 = b2 + c2 − 2bccosA

b

c

a

BA

C

Using the Law of Cosines (SAS)

Find b to the nearest tenth.

b2 = a2 + c2 − 2accosB Law of Cosines

b2 = 222 + 102 − 2(22)(10)cos44 Substitute

b 16.35513644

b 16.4

 

 

10

4422

b

CB

A

b2 = a2 + c2 − 2accosB Law of Cosines

4.42 = 6.72 + 7.12 − 2(6.7)(7.1)cosV Substitute

Find V to the nearest tenth.

Using the Law of Cosines (SSS)

Solve for angle V.

19.36 = 44.89 + 50.41 − 95.14cosV Substitute

4.4

7.1

6.7

VU

T

Examples

14.5 m11.2 m

580

x

Law of Cosinesc2 = a2 + b2 − 2abcosC

x2 = 11.22 + 14.52 − 2(11.2)(14.5)cos580

x2 = 163.6

x = 12.8 m

xo

4

7

5

c2 = a2 + b2 − 2abcosC

42 = 52 + 72 − 2(5)(7)cosxo

16 = 25 + 49 − 70cosxo

-58 = − 70cosxo

.829 = cosxo

34o = xp.529: 1-4, 7-15

odd