Chapter 8Section 8.2

Law of Cosines

Law of Cosines

In any triangle (not necessarily a right triangle) the square of the length of one side of the triangle is equal to the sum of the squares of the two remaining sides minus twice the product of the cosine of the angle of the opposite side and the lengths of the two sides forming the angle. It is like the "Pythagorean Theorem" for any triangle.

B

A

c

b

a

C

The Law of Cosines enables you to solve the triangle in the cases where your know all 3 sides (SSS) or 2 sides and the included angle (SAS).

8

7

5

A

B

C

9

10

a

55 °B

CExample:Find the remaining sides and angles of the triangle pictured to the right (SAS).

To find a, apply the Law of Cosines:

To find C, apply the Law of Sines: Subtract o find angle B:

𝐵≈180 °−55 °−68.273 °=56.727 °In this case we do not need to worry about another angle C since this (SAS) always determines a unique triangle.

So now, given 3 sides, or a pair of sides and an angle, or a pair of angles and a side you can now determine all possible triangles.

Example:

A plane takes off from an airport and travels 15 miles north. It then turns and travel on a heading that is east of north for miles.

a) How far is the plane from the airport?

b) How many degrees east of north is the plane from the airport?

Draw a picture.

Airport

15

20

25°

155°

To find a, apply the Law of Cosines:

a

B

To find B, apply the Law of Sines: