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:

Show Answer
As we can identify the 2 angles as 52° and 71° resepctively, we can solve for the value of the missing angle:

\(\theta = 180° - (52° + 71°)\)

\(\theta = 180° - 123°\)

\(\theta = 57°\)
Once we solve for the missing angle, we can use the sin law formula to solve for x:

\(\cfrac{\textcolor{red}{x}}{\textcolor{crimson}{\sin 52°}} = \cfrac{\textcolor{green}{8}}{\textcolor{seagreen}{\sin 57°}}\)

\(\cfrac{\textcolor{red}{x}}{\textcolor{crimson}{\sin 52°}} = 9.5389\)

\(\textcolor{red}{x} = (9.5389)(0.788)\)

\(\textcolor{red}{x = 7.5167}\)
Therefore, we can determine that \(\textcolor{red}{x = 7.516}\).

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 \(m = 23\;[cm]\).

Show Answer
First, we can draw the triangle to help visualize the problem:

We can use the Law of Sines to solve for \(\textcolor{seagreen}{\angle D}\):

\(\cfrac{\textcolor{crimson}{\sin C}}{\textcolor{red}{c}} = \cfrac{\textcolor{seagreen}{\sin D}}{\textcolor{green}{d}} = \cfrac{\textcolor{royalblue}{\sin M}}{\textcolor{blue}{m}}\)

\(\cfrac{\textcolor{royalblue}{\sin M}}{\textcolor{blue}{23}} = \cfrac{\textcolor{seagreen}{\sin 55°}}{\textcolor{green}{21}}\)

\((\cancel{\textcolor{blue}{23}})(\cfrac{\textcolor{royalblue}{\sin M}}{\cancel{\textcolor{blue}{23}}}) = (\textcolor{blue}{23})(\cfrac{\textcolor{seagreen}{\sin 55°}}{\textcolor{green}{21}}\))

\(\textcolor{royalblue}{\sin M} = \cfrac{(\textcolor{seagreen}{\sin 55°})(\textcolor{blue}{23})}{\textcolor{green}{21}}\)

\(\textcolor{royalblue}{\angle M} = \sin⁻¹(0.897166524)\)

\(\textcolor{royalblue}{\angle M = 63.78° = 64°}\)
We can now use the Triangle Sum Theorem to solve for \(\textcolor{crimson}{\angle C}\). We can do this by subtracting the other 2 angles, \(\textcolor{seagreen}{\angle D}\) and \(\textcolor{royalblue}{\angle M}\) from 180°:

\(\textcolor{crimson}{\angle C} = 180° - (\textcolor{seagreen}{\angle D} + \textcolor{royalblue}{\angle M})\)

\(\textcolor{crimson}{\angle C} = 180° - (\textcolor{seagreen}{55°} + \textcolor{royalblue}{64°})\)

\(\textcolor{crimson}{\angle C} = 180° - 119°\)

\(\textcolor{crimson}{\angle C = 61°}\)
Finally, we can use the Law of Sines to solve for \(\textcolor{red}{c}\):

\(\cfrac{\textcolor{red}{c}}{\textcolor{crimson}{\sin 61°}} = \cfrac{\textcolor{green}{21}}{\textcolor{seagreen}{\sin 55°}}\)

\((\textcolor{crimson}{\sin 61°})(\cfrac{\textcolor{red}{c}}{\textcolor{crimson}{\sin 61°}}) = (\textcolor{crimson}{\sin 61°})(\cfrac{\textcolor{green}{21}}{\textcolor{seagreen}{\sin 55°}})\)

\(\textcolor{red}{c} = \cfrac{(\textcolor{crimson}{\sin 61°})(\textcolor{green}{21})}{\textcolor{seagreen}{\sin 55°}}\)

\(\textcolor{red}{c = 22.42\;[cm]}\)
Therefore, we can determine that \(\textcolor{red}{c = 22.42\;[cm]}\), \(\textcolor{crimson}{\angle C = 61°}\) and \(\textcolor{royalblue}{\angle M = 64°}\)

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.

Show Answer
i. We can use the Law of Sines to determine the value of \(\textcolor{crimson}{\angle X}\):

\(\cfrac{\textcolor{crimson}{\sin X}}{\textcolor{red}{x}} = \cfrac{\textcolor{royalblue}{\sin Z}}{\textcolor{blue}{z}}\)

\(\cfrac{\textcolor{crimson}{\sin X}}{\textcolor{red}{13}} = \cfrac{\textcolor{royalblue}{\sin 52}}{\textcolor{blue}{15}}\)

\((\cancel{\textcolor{red}{13}})(\cfrac{\textcolor{crimson}{\sin X}}{\cancel{\textcolor{red}{13}}}) = (\textcolor{red}{13})(\cfrac{\textcolor{royalblue}{\sin 52}}{\textcolor{blue}{15}})\)

\(\textcolor{crimson}{\sin X} = \cfrac{(0.788)(13)}{\textcolor{blue}{15}}\)

\(\textcolor{crimson}{\angle X} = \sin⁻¹(0.682942653)\)

\(\textcolor{crimson}{\angle X = 43°}\)
Therefore, we can determine that \(\textcolor{crimson}{\angle X = 43°}\).

ii. In order to determine the length of the fence, we first need to determine its corresponding angle \(\textcolor{seagreen}{\angle Y}\). We can do this by subtracting the other 2 angles, \(\textcolor{crimson}{\angle X}\) and \(\textcolor{royalblue}{\angle Z}\) from 180°:

\(\textcolor{seagreen}{\angle Y} = 180° - (\textcolor{crimson}{\angle X} + \textcolor{royalblue}{\angle Z})\)

\(\textcolor{seagreen}{\angle Y} = 180° - (43° + 52°)\)

\(\textcolor{seagreen}{\angle Y} = 180° - 95°\)

\(\textcolor{seagreen}{\angle Y = 85°}\)
We can now use the Law of Sines to determine the length of the fence. In this instance, \(\textcolor{green}{y}\) will represent the length:

\(\cfrac{\textcolor{green}{y}}{\textcolor{seagreen}{\sin Y}} = \cfrac{\textcolor{blue}{z}}{\textcolor{royalblue}{\sin Z}}\)

\(\cfrac{\textcolor{green}{y}}{\textcolor{seagreen}{\sin 85°}} = \cfrac{\textcolor{blue}{15}}{\textcolor{royalblue}{\sin 52°}}\)

\((\cancel{\textcolor{seagreen}{\sin 85°}})(\cfrac{\textcolor{green}{y}}{\cancel{\textcolor{seagreen}{\sin 85°}}}) = (\textcolor{seagreen}{\sin 85°})(\cfrac{\textcolor{blue}{15}}{0.788})\)

\(\textcolor{green}{y} = \cfrac{(\textcolor{blue}{15})({0.996})}{0.788}\)

\(\textcolor{green}{y = 18.96\;[m] = 19\;[m]}\)
Therefore, we can determine that the length of the fence is 19m.