## How to Use the Law of Sines

- Label the angles \(A\), \(B\), \(C\) and the sides opposite the angles \(a\), \(b\), and \(c\).
- Identify which measurements you know and which one you are looking for.
- Plug the numbers into the Law of Sines Formula, cross multiply, and solve.

## Examples

Find the value of x.

*Which sides and angles are opposite of each other?*

*Which measurements are given by the problem? *

The diagram tells me that…

\[\red A=30^\circ\]

\[\red a = 5\]

\[\yellow c = 4\]

*Which measurement are you looking for?*

I am looking for \(\purple x\) which is the measurement of the angle opposite of side \(\purple c\), so…

\[\yellow C = x\]

*Which version of the Law of Sines Formula will you use?*

I know \(\purple A\), \(\purple a\), \(\purple c\), and I am looking for \(\purple c\). So, I will use this version of the Law of Sines Formula…

\[\frac{a}{sin \, A}=\frac{c}{sin \, C}\]

When I replace \(\purple A=30^\circ\), \(\purple a=5\), \(\purple c=4\), and \(\purple C=x\) into the formula, I get…

\[\frac{\purple 5}{sin \, {\purple 30^\circ}}=\frac{\purple 4}{sin \, {\purple x}}\]

To solve for x, I will first cross-multiply the fractions.

\[{\purple 5}sin{\purple x}={\purple 4}sin{\purple 30^\circ}\]

The unit circle tells me that the sine of \(30^\circ\) is \(0.5\) and when I multiply that by \(4\), I get…

\[{\purple 5}sin{\purple x}=2\]

Then I can divide both sides of the equation by \(\purple 5\).

\[sin{\purple x}=0.4\]

When I take the arcsine of both sides, my calculator tells me that…

\[{\purple x}=27.6^\circ\]