You and a group of your friends are out in the country playing paintball. You have a playing field that is a triangle with sides of 50 meters, 50 meters, and 80 meters. You're trying to figure out what the angle is between the side that has a length of 50 meters and the other side that has a length of 50 meters. Is there a way to do this?

### Using Law of Cosines to Find Unknown Angles

The Law of Cosines is a natural extension of the Pythagorean Theorem that allows us to perform calculations to find the sides of triangles that are oblique.

Remember that the Law of Cosines is a generalization of the Pythagorean Theorem, where the angle \begin{align*}C\end{align*} is the angle between the two given sides of a triangle:

\begin{align*}c^2 = a^2 + b^2 - 2(a)(b)\cos C\end{align*}

You'll notice that if this were a right triangle, \begin{align*}\cos C = \cos 90^\circ = 0\end{align*}, and so the third term would disappear, leaving the familiar Pythagorean Theorem.

Another situation where we can apply the Law of Cosines is when we know all three sides in a triangle (SSS) and we need to find one of the angles. The Law of Cosines allows us to find any of the three angles in the triangle. First, we will look at how to apply the Law of Cosines in this case, and then we will look at the real-world application given in the problem above.

#### Using the Law of Cosines

For the following problems, solve using the Law of Cosines.

1. An architect is designing a kitchen for a client. When designing a kitchen, the architect must pay special attention to the placement of the stove, sink, and refrigerator. In order for a kitchen to be utilized effectively, these three amenities must form a triangle with each other. This is known as the “work triangle.” By design, the three parts of the work triangle must be *no less than 3 feet apart and no more than 7 feet apart*. Based on the dimensions of the current kitchen, the architect has determined that the sink will be 3.6 feet away from the stove and 5.7 feet away from the refrigerator. The sink forms a \begin{align*}103^\circ\end{align*} angle with the stove and the refrigerator. If the architect moves the stove so that it is 4.2 feet from the sink and makes the fridge 6.8 feet from the stove, how does this affect the angle the sink forms with the stove and the refrigerator?

In order to find how the angle is affected, we will again need to use the Law of Cosines, but because we do not know the measures of any of the angles, we solve for \begin{align*}Y\end{align*}.

\begin{align*}6.8^2 & = 4.2^2 + 5.7^2 - 2(4.2)(5.7) \cos Y && \text{Law of Cosines} \\ 46.24 & = 17.64 + 32.49 - 2(4.2)(5.7) \cos Y && \text{Simplify squares} \\ 46.24 & = 17.64 + 32.49 - 47.88 \cos Y && \text{Multiply} \\ 46.24 & = 50.13 - 47.88 \cos Y && \text{Add} \\ - 3.89 & = -47.88 \cos Y && \text{Subtract} \\ 0.0812447786 & = \cos Y && \text{Divide} \\ 85.3^\circ & \approx Y && \cos^{-1} \ (0.081244786)\end{align*}

The new angle would be \begin{align*}85.3^\circ\end{align*}, which means it would be \begin{align*}17.7^\circ\end{align*} less than the original angle.

2. In oblique \begin{align*}\triangle{MNO}, m = 45, n = 28\end{align*}, and \begin{align*}o = 49\end{align*}. Find \begin{align*}\angle{M}\end{align*}.

Since we know all three sides of the triangle, we can use the Law of Cosines to find \begin{align*}\angle{M}\end{align*}.

\begin{align*}45^2 & = 28^2 + 49^2 - 2(28)(49) \cos M && \text{Law of Cosines} \\ 2025 & = 784 + 2401 - 2(28)(49) \cos M && \text{Simplify squares} \\ 2025 & = 784 + 2401 - 2744 \cos M && \text{Multiply} \\ 2025 & = 3185 - 2744 \cos M && \text{Add} \\ -1160 & = -2744 \cos M && \text{Subtract}\ 3185 \\ 0.422740525 & = \cos M && \text{Divide by}\ -2744 \\ 65^\circ & \approx M && \cos^{-1} \ (0.422740525) \end{align*}

It is important to note that we could use the Law of Cosines to find \begin{align*}\angle{N}\end{align*} or \begin{align*}\angle{O}\end{align*} also.

3. Sam is building a retaining wall for a garden that he plans on putting in the back corner of his yard. Due to the placement of some trees, the dimensions of his wall need to be as follows: side \begin{align*}1 = 12ft\end{align*}, side \begin{align*}2 = 18ft\end{align*}, and side \begin{align*}3 = 22ft\end{align*}. At what angle do side 1 and side 2 need to be? Side 2 and side 3? Side 1 and side 3?

Since we know the measures of all three sides of the retaining wall, we can use the Law of Cosines to find the measures of the angles formed by adjacent walls. We will refer to the angle formed by side 1 and side 2 as \begin{align*}\angle{A}\end{align*}, the angle formed by side 2 and side 3 as \begin{align*}\angle{B}\end{align*}, and the angle formed by side 1 and side 3 as \begin{align*}\angle{C}\end{align*}. First, we will find \begin{align*}\angle{A}\end{align*}.

\begin{align*}22^2 & = 12^2 + 18^2 - 2(12)(18) \cos A && \text{Law of Cosines} \\ 484 & = 144 + 324 - 2(12)(18) \cos A && \text{Simplify squares} \\ 484 & = 144 + 324 - 432 \cos A && \text{Multiply} \\ 484 & = 468 - 432 \cos A && \text{Add} \\ 16 & = -432 \cos A && \text{Subtract}\ 468 \\ -0.037037037 & \approx \cos A && \text{Divide by}\ -432 \\ 92.1^\circ & \approx A && \cos^{-1} \ (-0.037037037) \end{align*}

Next we will find the measure of \begin{align*}\angle{B}\end{align*} also by using the Law of Cosines.

\begin{align*}18^2 & = 12^2 + 22^2 - 2(12)(22) \cos B && \text{Law of Cosines} \\ 324 & = 144 + 484 - 2(12)(22) \cos B && \text{Simplify squares} \\ 324 & = 144 + 484 - 528 \cos B && \text{Multiply} \\ 324 & = 628 - 528 \cos B && \text{Add} \\ -304 & = -528 \cos B && \text{Subtract}\ 628 \\ 0.575757576 & = \cos B && \text{Divide by}\ -528 \\ 54.8^\circ & \approx B && \cos^{-1} \ (0.575757576) \end{align*}

Now that we know two of the angles, we can find the third angle using the Triangle Sum Theorem, \begin{align*}\angle{C} = 180 - (92.1 + 54.8) = 33.1^\circ\end{align*}.

### Examples

#### Example 1

Earlier, you were asked to figure out what the angle is of a triangle.

You can use the Law of Cosines to solve this problem:

\begin{align*} c^2 = a^2 + b^2 + 2ab\cos \theta\\ 80^2 = 50^2 + 50^2 + (2)(50)(50)\cos \theta\\ 80^2 - 50^2 - 50^2 = (2)(50)(50)\cos \theta\\ 6400 - 2500 - 2500 = (2)(50)(50)\cos \theta\\ 1400 = (2)(50)(50)\cos \theta\\ \cos \theta = \frac{1400}{5000}\\ \theta = \cos^{-1} (.28)\\ \theta = 73.74^\circ\\ \end{align*}

The angle in your paintball course is rather large, measuring \begin{align*}73.74^\circ\end{align*}

#### Example 2

Find the largest angle in the triangle below, where \begin{align*}t = 6, r = 7, i = 11\end{align*}

\begin{align*}11^2 = 6^2 + 7^2 - 2 \cdot 6 \cdot 7 \cdot \cos I, \angle{I} \approx 115.4^\circ\end{align*}

#### Example 3

Find the smallest angle in the triangle below, where \begin{align*}q = 17, d = 12.8, r = 18.6, \angle{Q} = 62.4^\circ\end{align*}

\begin{align*}12.8^2 = 17^2 + 18.6^2 - 2 \cdot 17 \cdot 18.6 \cdot \cos D, \angle{D} \approx 41.8^\circ\end{align*}

#### Example 4

Find the second largest angle in the triangle below, where \begin{align*}c = 9, d = 11, m = 13\end{align*}

\begin{align*}11^2 = 9^2 + 13^3 - 2 \cdot 9 \cdot 13 \cdot \cos D, \angle{D} \approx 56.5^\circ\end{align*}

### Review

- If you know the lengths of all three sides of a triangle, how can you identify the smallest angle of the triangle? The largest angle?
- If you know the measures of two angles of a triangle, how can you find the measure of the third angle?

Use the triangle below to answer questions 3-5.

- What is the measure of the smallest angle of the triangle?
- What is the measure of the largest angle of the triangle?
- What is the measure of the third angle of the triangle?

Use the triangle below to answer questions 6-8.

- What is the measure of the smallest angle of the triangle?
- What is the measure of the largest angle of the triangle?
- What is the measure of the third angle of the triangle?

Use the triangle below to answer questions 9-11.

- What is the measure of the smallest angle of the triangle?
- What is the measure of the largest angle of the triangle?
- What is the measure of the third angle of the triangle?

Use the triangle below to answer questions 12-14.

- What is the measure of the smallest angle of the triangle?
- What is the measure of the largest angle of the triangle?
- What is the measure of the third angle of the triangle?

Use the triangle below to answer questions 15-17.

- What is the measure of the smallest angle of the triangle?
- What is the measure of the largest angle of the triangle?
- What is the measure of the third angle of the triangle?

Use the triangle below to answer questions 18-20.

- What is the measure of the smallest angle of the triangle?
- What is the measure of the largest angle of the triangle?
- What is the measure of the third angle of the triangle?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 5.2.