# ### Law of Sines

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## How to Use the Law of Sines

1. Label the angles $$A$$, $$B$$, $$C$$ and the sides opposite the angles $$a$$, $$b$$, and $$c$$.
2. Identify which measurements you know and which one you are looking for.
3. 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…

${\purple x}=27.6^\circ$