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:

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:

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

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:

\(\textcolor{red}{d}^2 = (\textcolor{green}{11})^2 + (\textcolor{blue}{14})^2 - 2(\textcolor{green}{11})(\textcolor{blue}{14})\textcolor{crimson}{\cos(82°)}\)

\(\textcolor{red}{d}^2 = 121 + 196 - (308)(0.139)\)

\(\textcolor{red}{d}^2 = 317 - 42.87\)

\(\sqrt{\textcolor{red}{d}^2} = \sqrt{274.13}\)

\(\textcolor{red}{d = 16.55\;[cm] = 16.6\;[cm]}\)

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.

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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:

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

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

\(\textcolor{royalblue}{\cos P} = \cfrac{20.2}{38.76}\)

\(\textcolor{royalblue}{\angle P} = \cos⁻¹(0.52115583)\)

\(\textcolor{royalblue}{\angle P} = 58.59° = 59°\)

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 \(v = 4.0\;[cm]\).

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

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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

i. Find the interior angles of the reflecting pool

ii. Find the surface area of the water in the refelcting pool

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