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:
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:
Determine the missing side length of the following triangle to 2 decimal places:
Since we can identify \(\textcolor{crimson}{\angle \text{A} = 60°}\), \(\textcolor{royalblue}{\angle \text{C} = 50°}\) and \(\textcolor{blue}{c = 10}\), we can solve for \(\textcolor{red}{a}\) using the Law of Sines:
\(\cfrac{\textcolor{red}{a}}{\textcolor{crimson}{\sin60°}} = \cfrac{\textcolor{blue}{c}}{\textcolor{royalblue}{\sin50°}}\)
\(\cfrac{\textcolor{red}{a}}{\textcolor{crimson}{\sin60°}} = 7.66\)
\(\textcolor{red}{a} = (7.66)(\textcolor{crimson}{\sin60})\)
\(\textcolor{red}{a} = (7.66)(0.866)\)
\(a = 6.6337 \approx 6.63\)
Therefore, we can determine that \(\boldsymbol{a \approx 6.63}\).
As we can identify the 2 angles as \(52°\) and \(71°\) resepctively, we can solve for the value of the missing angle:
\(\theta = 180° - (52° + 71°)\)
\(\theta = 180° - 123°\)
\(\theta = 57°\)
Since we solved for the missing angle, we can use the Sine Law formula to solve for \(x\):
\(\cfrac{\textcolor{red}{x}}{\textcolor{crimson}{\sin 52°}} = \cfrac{\textcolor{green}{8}}{\textcolor{seagreen}{\sin 57°}}\)
\(\cfrac{\textcolor{red}{x}}{\textcolor{crimson}{\sin 52°}} = 9.5389\)
\(\textcolor{red}{x} = (9.5389)(0.788)\)
\(x = 7.5167 \approx 7.52\)
Therefore, we can determine that \(\boldsymbol{x \approx 7.52}\).
In order to solve for angles more effectively, the Law of Sines can alternatively be expressed as:
Determine the missing angle of the following triangle to the nearest degree:
Since we are solving for an angle, we are using the alternate version of the Law of Sines:
\(\cfrac{\textcolor{royalblue}{\sin \text{R}}}{\textcolor{blue}{r}} = \cfrac{\textcolor{crimson}{\sin \text{P}}}{\textcolor{red}{p}}\)
\(\cfrac{\textcolor{royalblue}{\sin \text{R}}}{\textcolor{blue}{4.1}} = \cfrac{\textcolor{crimson}{\sin64°}}{\textcolor{red}{5.7}}\)
\((\cancel{\textcolor{blue}{4.1}})\left(\cfrac{\textcolor{royalblue}{\sin \text{R}}}{\cancel{\textcolor{blue}{4.1}}}\right) = (\textcolor{blue}{4.1})\left(\cfrac{0.899}{\textcolor{red}{5.7}}\right)\)
\(\textcolor{royalblue}{\sin \text{R}} = \cfrac{(\textcolor{royalblue}{4.1})(0.899)}{\textcolor{red}{5.7}}\)
\(\textcolor{royalblue}{\angle \text{R}} = \sin⁻¹(0.64650098)\)
\(\angle \text{R} = 40.27° \approx 40°\)
Therefore, we can determine that \(\boldsymbol{\angle \textbf{R} \approx 40°}\).
For \(\triangle \text{DMC}\), where \(\angle \text{D} = 55°\), \(d = 21\;[\text{cm}]\), and \(m = 23\;[\text{cm}]\):
i. We can draw the triangle as such to help visualize the problem:
ii. We can use the Law of Sines to solve for \(\textcolor{royalblue}{\angle \text{M}}\):
\(\cfrac{\textcolor{crimson}{\sin \text{C}}}{\textcolor{red}{c}} = \cfrac{\textcolor{seagreen}{\sin \text{D}}}{\textcolor{green}{d}} = \cfrac{\textcolor{royalblue}{\sin \text{M}}}{\textcolor{blue}{m}}\)
\(\cfrac{\textcolor{royalblue}{\sin \text{M}}}{\textcolor{blue}{23}} = \cfrac{\textcolor{seagreen}{\sin 55°}}{\textcolor{green}{21}}\)
\((\cancel{23})\left(\cfrac{\textcolor{royalblue}{\sin \text{M}}}{\cancel{\textcolor{blue}{23}}}\right) = (23)\left(\cfrac{\textcolor{seagreen}{\sin 55°}}{\textcolor{green}{21}}\right)\)
\(\textcolor{royalblue}{\sin \text{M}} = \cfrac{(\textcolor{seagreen}{\sin 55°})(\textcolor{blue}{23})}{\textcolor{green}{21}}\)
\(\textcolor{royalblue}{\angle \text{M}} = \sin⁻¹(0.897166524)\)
\(\textcolor{royalblue}{\angle \text{M} = 63.78° \approx 64°}\)
We can now use the Triangle Sum Theorem to solve for \(\textcolor{crimson}{\angle \text{C}}\). We can do this by subtracting the other \(2\) angles, \(\textcolor{seagreen}{\angle \text{D}}\) and \(\textcolor{royalblue}{\angle \text{M}}\) from \(180°\):
\(\textcolor{crimson}{\angle \text{C}} = 180° - (\textcolor{seagreen}{\angle \text{D}} + \textcolor{royalblue}{\angle \text{M}})\)
\(\textcolor{crimson}{\angle \text{C}} = 180° - (\textcolor{seagreen}{55°} + \textcolor{royalblue}{64°})\)
\(\textcolor{crimson}{\angle \text{C}} = 180° - 119°\)
\(\textcolor{crimson}{\angle \text{C} = 61°}\)
Finally, we can use the Law of Sines to solve for \(\textcolor{red}{c}\):
\(\cfrac{\textcolor{red}{c}}{\textcolor{crimson}{\sin 61°}} = \cfrac{\textcolor{green}{21}}{\textcolor{seagreen}{\sin 55°}}\)
\((\cancel{\sin 61°})\left(\cfrac{\textcolor{red}{c}}{\cancel{\textcolor{crimson}{\sin 61°}}}\right) = (\sin 61°)\left(\cfrac{\textcolor{green}{21}}{\textcolor{seagreen}{\sin 55°}}\right)\)
\(\textcolor{red}{c} = \cfrac{(\textcolor{crimson}{\sin 61°})(\textcolor{green}{21})}{\textcolor{seagreen}{\sin 55°}}\)
\(\textcolor{red}{c = 22.42\;[\text{cm}]}\)
Therefore, we can determine that \(\boldsymbol{\textcolor{red}{c = 22.42\;[\textbf{cm}]}}\), \(\boldsymbol{\textcolor{crimson}{\angle \textbf{C} = 61°}}\) and \(\boldsymbol{\textcolor{royalblue}{\angle \textbf{M} = 64°}}\).
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. We can use the Law of Sines to determine the value of \(\textcolor{crimson}{\angle \text{X}}\):
\(\cfrac{\textcolor{crimson}{\sin \text{X}}}{\textcolor{red}{x}} = \cfrac{\textcolor{royalblue}{\sin \text{Z}}}{\textcolor{blue}{z}}\)
\(\cfrac{\textcolor{crimson}{\sin \text{X}}}{\textcolor{red}{13}} = \cfrac{\textcolor{royalblue}{\sin 52}}{\textcolor{blue}{15}}\)
\((\cancel{\textcolor{red}{13}})\left(\cfrac{\textcolor{crimson}{\sin \text{X}}}{\cancel{\textcolor{red}{13}}}\right) = (\textcolor{red}{13})\left(\cfrac{\textcolor{royalblue}{\sin 52}}{\textcolor{blue}{15}}\right)\)
\(\textcolor{crimson}{\sin \text{X}} = \cfrac{(0.788)(13)}{\textcolor{blue}{15}}\)
\(\textcolor{crimson}{\angle \text{X}} = \sin^{-1}(0.682942653)\)
\(\textcolor{crimson}{\angle \text{X} = 43.07^{\circ} \approx 43°}\)
Therefore, we can determine that \(\textcolor{crimson}{\boldsymbol{\angle \textbf{X} = 43°}}\).
ii. In order to determine the length of the fence, we first need to determine its corresponding angle \(\textcolor{seagreen}{\angle \text{Y}}\). We can do this by subtracting the other 2 angles, \(\textcolor{crimson}{\angle \text{X}}\) and \(\textcolor{royalblue}{\angle \text{Z}}\) from \(180°\):
\(\textcolor{seagreen}{\angle \text{Y}} = 180° - (\textcolor{crimson}{\angle \text{X}} + \textcolor{royalblue}{\angle \text{Z}})\)
\(\textcolor{seagreen}{\angle \text{Y}} = 180° - (43° + 52°)\)
\(\textcolor{seagreen}{\angle \text{Y}} = 180° - 95°\)
\(\textcolor{seagreen}{\angle \text{Y} = 85°}\)
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:
\(\cfrac{\textcolor{green}{y}}{\textcolor{seagreen}{\sin \text{Y}}} = \cfrac{\textcolor{blue}{z}}{\textcolor{royalblue}{\sin \text{Z}}}\)
\(\cfrac{\textcolor{green}{y}}{\textcolor{seagreen}{\sin 85°}} = \cfrac{\textcolor{blue}{15}}{\textcolor{royalblue}{\sin 52°}}\)
\((\cancel{\textcolor{seagreen}{\sin 85°}})\left(\cfrac{\textcolor{green}{y}}{\cancel{\textcolor{seagreen}{\sin 85°}}}\right) = (\textcolor{seagreen}{\sin 85°})\left(\cfrac{\textcolor{blue}{15}}{0.788}\right)\)
\(\textcolor{green}{y} = \cfrac{(\textcolor{blue}{15})({0.996})}{0.788}\)
\(\textcolor{green}{y = 18.96\;[\text{m}] \approx 19\;[\text{m}]}\)
Therefore, we can determine that the length of the fence is \(\boldsymbol{\textcolor{green}{19 \; [\text{m}]}}\).