Toby draws a triangle that has two side lengths of 8 inches and 5 inches. He measures the included angle with a protractor and gets

### Guidance

The Law of Cosines can be used to solve for the third side of a triangle when two sides and the included angle are known in a triangle. consider the non right triangle below in which we know

Now we can use the Pythagorean Theorem to relate the lengths of the segments in each of the right triangles shown.

Triangle 1:

Triangle 2:

Since both equations are equal to

Recall that we know the values of

Finally, we can replace

Keep in mind that

#### Example A

Find

**Solution:** Replacing the variables in the formula with the given information and solve for

#### Example B

Find

**Solution:** This time we are given the sides surrounding angle

#### Example C

Rae is making a triangular flower garden. One side is bounded by her porch and a second side is bounded by her fence. She plans to put in a stone border on the third side. If the length of the porch is 10 ft and the length of the fence is 15 ft and they meet at a

**Solution:** Let the two known side lengths be

So Rae will need to create a 19.4 ft stone border.

**Concept Problem Revisit** We are trying to find

**Solution:** Replacing the variables in the formula with the given information and solve for

Therefore, the third side is approximately 8.26 inches long.

### Guided Practice

1. Find \begin{align*}c\end{align*} when \begin{align*}m \angle C=75^\circ, a = 32\end{align*} and \begin{align*}b = 40\end{align*}.

2. Find \begin{align*}b\end{align*} when \begin{align*}m \angle B=120^\circ, a = 11\end{align*} and \begin{align*}c =17\end{align*}.

3. Dan likes to swim laps across a small lake near his home. He swims from a pier on the north side to a pier on the south side multiple times for a workout. One day he decided to determine the length of his swim. He determines the distances from each of the piers to a point on land and the angles between the piers from that point to be \begin{align*}50^\circ\end{align*}. How many laps does Dan need to swim to cover 1000 meters?

#### Answers

1. \begin{align*}c^2 &=32^2+40^2-2(32)(40) \cos 75^\circ \\ c^2 & \approx 1961.42 \\ c & \approx 44.3\end{align*}

2. \begin{align*}b^2 &=11^2+17^2-2(11)(17) \cos 120^\circ \\ b^2 & \approx 597 \\ b & \approx 24.4\end{align*}

3. \begin{align*}c^2 &=30^2+35^2-2(30)(35) \cos 50^\circ \\ c^2 & \approx 775.146 \\ c & \approx 27.84\end{align*}

Since each lap is 27.84 meters, Dan must swim \begin{align*}\frac{1000}{27.84} \approx 36\end{align*} laps.

### Explore More

Use the Law of Cosines to find the value of \begin{align*}x\end{align*}, to the nearest tenth, in problems 1 through 6.

For problems 7 through 10, find the unknown side of the triangle. Round your answers to the nearest tenth.

- Find \begin{align*}c\end{align*}, given \begin{align*}m \angle C=105^\circ\end{align*}, \begin{align*}a = 55\end{align*} and \begin{align*}b = 61\end{align*}.
- Find \begin{align*}b\end{align*}, given \begin{align*}m \angle B=26^\circ\end{align*}, \begin{align*}a = 33\end{align*} and \begin{align*}c = 24\end{align*}.
- Find \begin{align*}a\end{align*}, given \begin{align*}m \angle A=77^\circ\end{align*}, \begin{align*}b = 12\end{align*} and \begin{align*}c = 19\end{align*}.
- Find \begin{align*}b\end{align*}, given \begin{align*}m \angle B=95^\circ\end{align*}, \begin{align*}a = 28\end{align*} and \begin{align*}c = 13\end{align*}.
- Explain why when \begin{align*}m \angle C=90^\circ\end{align*}, the Law of Cosines becomes the Pythagorean Theorem.
- Luis is designing a triangular patio in his backyard. One side, 20 ft long, will be up against the side of his house. A second side is bordered by his wooden fence. If the fence and the house meet at a \begin{align*}120^\circ\end{align*} angle and the fence is 15 ft long, how long is the third side of the patio?