# Law of Sines

The Law of Sines is a way of solving for the missing angles and side lengths for any oblique (non-right) triangle.

This can be expressed using the formula:

$$\cfrac{\textcolor{red}{a}}{\textcolor{crimson}{\sin{A}}} = \cfrac{\textcolor{green}{b}}{\textcolor{seagreen}{\sin{B}}} = \cfrac{\textcolor{blue}{c}}{\textcolor{royalblue}{\sin{C}}}$$

This formula indicates the ratios of the sides divided by the sine value of their corresponding angle are all the same. It is mainly used to solve for side lengths.

In order to use the Law of Sines, you need to know either:

• 2 angles and a side length opposite one of them:
• 2 side lengths and an angle opposite one of them:
Example

Determine the missing side length of the following triangle:

Since we can identify $$\textcolor{crimson}{\angle A = 60°}$$, $$\textcolor{royalblue}{\angle C = 50°}$$ and $$\textcolor{blue}{c = 10}$$, we can solve for $$\textcolor{red}{a}$$ using the Law of Sines:

$$\cfrac{\textcolor{red}{a}}{\textcolor{crimson}{\sin60°}} = \cfrac{\textcolor{blue}{c}}{\textcolor{royalblue}{\sin50°}}$$

$$\cfrac{\textcolor{red}{a}}{\textcolor{crimson}{\sin60°}} = 7.66$$

$$\textcolor{red}{a} = (7.66)(\textcolor{crimson}{\sin60})$$
$$\textcolor{red}{a} = (7.66)(0.866)$$
$$\textcolor{red}{a = 6.6337}$$

Therefore, we can determine that $$\textcolor{red}{a = 6.6337}$$.

Determine the missing side length of the following triangle:

In order to solve for angles more effectively, the Law of Sines can alternatively be expressed as:

$$\cfrac{\textcolor{crimson}{\sin A}}{\textcolor{red}{a}} = \cfrac{\textcolor{seagreen}{\sin B}}{\textcolor{green}{b}} = \cfrac{\textcolor{royalblue}{\sin C}}{\textcolor{blue}{c}}$$

Example

Determine the missing angle of the following triangle:

Since we are solving for angle, we are using the alternate version of the Law of Sines:

$$\cfrac{\textcolor{royalblue}{\sin R}}{\textcolor{blue}{r}} = \cfrac{\textcolor{crimson}{\sin P}}{\textcolor{red}{p}}$$

$$\cfrac{\textcolor{royalblue}{\sin R}}{\textcolor{blue}{4.1}} = \cfrac{\textcolor{crimson}{\sin64°}}{\textcolor{red}{5.7}}$$

$$(\cancel{\textcolor{blue}{4.1}})(\cfrac{\textcolor{royalblue}{\sin R}}{\cancel{\textcolor{blue}{4.1}}}) = (\textcolor{blue}{4.1})(\cfrac{0.899}{\textcolor{red}{5.7}})$$

$$\textcolor{royalblue}{\sin R} = \cfrac{(\textcolor{royalblue}{4.1})(0.899)}{\textcolor{red}{5.7}}$$

$$\textcolor{royalblue}{\angle R} = \sin⁻¹(0.64650098)$$
$$\textcolor{royalblue}{\angle R = 40.27° = 40°}$$

Therefore, we can determine that $$\textcolor{royalblue}{\angle R = 40°}$$.

Draw a diagram and label the given information. Then solve the triangle. In $$\triangle DMC$$, $$\angle D = 55°$$, $$d = 21\;[cm]$$, and $$m = 23\;[cm]$$.

Angela is building a garden in the shape of a triangle, as shown. She would like to put a fence on one side of the garden.
i. Find the angle formed by the fence and the side that is 15m.
ii. Find the length of the fence.