# Law of Cosines

The Law of Cosines is a way of solving for the missing angles and side lengths for any oblique (non-right angled) triangle. This method uses the formula:

$$\textcolor{blue}{c}^2 = \textcolor{red}{a}^2 + \textcolor{green}{b}^2 - 2\textcolor{red}{a}\textcolor{green}{b}\textcolor{royalblue}{\cos C}$$

This formula is used whenever we have 2 side lengths and an angle and need to the missing side length. It can also be rearranged as such:

$$\textcolor{royalblue}{\cos C} = \cfrac{\textcolor{red}{a}^2 + \textcolor{green}{b}^2 - \textcolor{blue}{c}^2}{2\textcolor{red}{a}\textcolor{green}{b}}$$

This formula is used whenever we have all 3 side lengths and need to determine an angle(s).

Example

Find the missing side length of the following triangle:

As we have 2 side lengths and an angle, we can use the side length formula, $$\textcolor{blue}{c}^2 = \textcolor{red}{a}^2 + \textcolor{green}{b}^2 - 2\textcolor{red}{a}\textcolor{green}{b}\textcolor{royalblue}{\cos(C)}$$, to determine the missing side length:

$$\textcolor{red}{d}^2 = \textcolor{green}{e}^2 + \textcolor{blue}{f}^2 - 2\textcolor{green}{e}\textcolor{blue}{f}\textcolor{crimson}{\cos D}$$
$$\textcolor{red}{d}^2 = (\textcolor{green}{11})^2 + (\textcolor{blue}{14})^2 - 2(\textcolor{green}{11})(\textcolor{blue}{14})\textcolor{crimson}{\cos(82°)}$$
$$\textcolor{red}{d}^2 = 121 + 196 - (308)(0.139)$$
$$\textcolor{red}{d}^2 = 317 - 42.87$$
$$\sqrt{\textcolor{red}{d}^2} = \sqrt{274.13}$$
$$\textcolor{red}{d = 16.55\;[cm] = 16.6\;[cm]}$$

Therefore, we can determine the length of $$\textcolor{red}{d}$$ is 16.6cm.

$$△MCB$$ has $$\angle M = 61°$$, $$c = 18\;[cm]$$, and $$b = 21\;[cm]$$. Sketch the triangle and and label the given information. Then, solve the triangle.

Example

Solve for the indicated angle, to the nearest degree:

As we have 3 side lengths, we can use the angle formula, $$\textcolor{royalblue}{\cos C} = \cfrac{\textcolor{red}{a}^2 + \textcolor{green}{b}^2 - \textcolor{blue}{c}^2}{2\textcolor{red}{a}\textcolor{green}{b}}$$, to solve for the missing angle:

$$\textcolor{royalblue}{\cos P} = \cfrac{\textcolor{red}{z}^2 + \textcolor{green}{j}^2 - \textcolor{blue}{p}^2}{2\textcolor{red}{z}\textcolor{green}{j}}$$

$$\textcolor{royalblue}{cosP} = \cfrac{(\textcolor{red}{5.1})^2 + (\textcolor{green}{3.8})^2 - (\textcolor{blue}{4.5})^2}{2(\textcolor{red}{5.1})(\textcolor{green}{3.8})}$$

$$\textcolor{royalblue}{\cos P} = \cfrac{26.01 + 14.44 - 20.25}{38.76}$$

$$\textcolor{royalblue}{\cos P} = \cfrac{20.2}{38.76}$$

$$\textcolor{royalblue}{\angle P} = \cos⁻¹(0.52115583)$$
$$\textcolor{royalblue}{\angle P} = 58.59° = 59°$$

Therefore, we can determine that the value of $$\textcolor{royalblue}{\angle P}$$ is 59°.

Sketch the triangle and and label the given information. Then, solve the triangle. In $$△MRV$$, $$m = 3.2\;[cm]$$, $$r = 3.5\;[cm]$$, and $$v = 4.0\;[cm]$$.

Find the length of the bridge, to the nearest metre.

Laurissa is designing a reflecting pool, in the shape of a triangle, for her backyard.
i. Find the interior angles of the reflecting pool
ii. Find the surface area of the water in the refelcting pool