The Law of Cosines is a generalized Pythagorean Theorem that allows you to solve for the missing sides and angles of a triangle even if it is not a right triangle. Suppose you have a triangle with sides 11, 12 and 13. What is the measure of the angle opposite the 11?

#### Watch This

http://www.youtube.com/watch?v=pGaDcOMdw48 Khan Academy: Law of Cosines

#### Guidance

The Law of Cosines is:

It is important to understand the proof:

You know four facts from the picture:

Once you verify for yourself that you agree with each of these facts, check algebraically that these next two facts must be true.

Now the Law of Cosines is ready to be proved using substitution, FOIL, more substitution and rewriting to get the order of terms right.

There are only two types of problems in which it is appropriate to use the Law of Cosines. The first is when you are given all three sides of a triangle and asked to find an unknown angle. This is called SSS like in geometry. The second situation where you will use the Law of Cosines is when you are given two sides and the included angle and you need to find the third side. This is called SAS.

**Example A**

Determine the measure of angle

**Solution:** It is necessary to set up the Law of Cosines equation very carefully with

**Example B**

Determine the length of side

**Solution:**

**Example C**

Determine the degree measure of angle

**Solution:** This problem must be done in two parts. First apply the Law of Cosines to determine the length of side

Now that you have all three sides you can apply the Law of Cosines again to find the unknown angle

**Concept Problem Revisited**

A triangle that has sides 11, 12 and 13 is not going to be a right triangle. In order to solve for the missing angle you need to use the Law of Cosines because this is a SSS situation.

#### Vocabulary

The * Law of Cosines* is a generalized Pythagorean Theorem that allows you to solve for the missing sides and angles of a triangle even if it is not a right triangle.

* SSS* refers to Side, Side, Side and refers to a property of congruent triangles in geometry. In this case it refers to the fact that all three sides are known in the problem.

* SAS* refers to Side, Angle, Side and refers to a property of congruent triangles in geometry. In this case it refers to the fact that the known quantities of a triangle are two sides and the included angle.

* Included angle* is the angle between two sides.

#### Guided Practice

1. Determine the length of side

2. Determine the measure of angle

3. Determine the measure of angle

**Answers:**

1.

2.

3. You could repeat the process from the previous question, or use the knowledge that the three angles in a triangle add up to 180.

#### Practice

For all problems, find angles in degrees rounded to one decimal place.

In

1. Find the measure of angle

2. Find the measure of angle

3. Find the measure of angle

4. Find the measure of angle

In

5. Find the measure of angle

6. Find the measure of angle

7. Find the measure of angle

In

8. Find the length of

9. Find the measure of angle

10. Find the measure of angle

11. Explain why the Law of Cosines is connected to the Pythagorean Theorem.

12. What are the two types of problems where you might use the Law of Cosines?

Determine whether or not each triangle is possible.

13.

14.

15.