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…
\[{\purple x}=27.6^\circ\]
