Sarine draws a triangle. She measures the length of the sides and records her measurements as follows. What is the measure of angle C of the triangle?

### Guidance

The Law of Cosines, , can be rearranged to facilitate the calculation of the measure of angle when and are all known lengths.

which can be further manipulated to .

#### Example A

Find the measure of the largest angle in the triangle with side lengths 12, 18 and 21.

**
Solution:
**
First, we must determine which angle will be the largest. Recall from Geometry that the longest side is opposite the largest angle. The longest side is 21 so we will let
since
is the angle we are trying to find. Let
and
and use the formula to solve for
as shown. It doesn’t matter which sides we assign to
and
. They are interchangeable in the formula.

**
Note:
**
Be careful to put parenthesis around the entire numerator and entire denominator on the calculator to ensure the proper order of operations. Your calculator screen should look like this:

#### Example B

Find the value of , to the nearest degree.

**
Solution:
**
The angle with measure
will be angle
so
and
. Remember,
and
are interchangeable in the formula. Now we can replace the variables with the known measures and solve.

#### Example C

Find the , if and .

**
Solution:
**
First, let’s rearrange the formula to reflect the sides given and requested angle:

, now plug in our values

**
Concept Problem Revisit
**

We can use the manipulated Law of Cosines to solve for C.

Therefore, the triangle is a right triangle.

### Guided Practice

1. Find the measure of in the diagram:

2. Find the measure of the smallest angle in the triangle with side lengths 47, 54 and 72.

3. Find , if and .

#### Answers

1.

2. The smallest angle will be opposite the side with length 47, so this will be our in the equation.

3. Rearrange the formula to solve for

### Explore More

Use the Law of Cosines to find the value of , to the nearest degree, in problems 1 through 6.

- Find the measure of the smallest angle in the triangle with side lengths 150, 165 and 200 meters.
- Find the measure of the largest angle in the triangle with side length 59, 83 and 100 yards.
- Find the if and .
- Find the if and .
- Find the if and .
- A triangular plot of land is bordered by a road, a fence and a creek. If the stretch along the road is 100 meters, the length of the fence is 115 meters and the side along the creek is 90 meters, at what angle do the fence and road meet?