Law of Sines

The law of sines of a triangle is a formula relating the angles of a triangle to their corresponding side lengths. It states the following.

First, let $ABC$ be a triangle.

Let $a=BC$, $b=AC$, and $c=AB$. Then, according to the law of sines: $$\frac {\sin \angle A}{a} = \frac {\sin \angle B}{b} = \frac {\sin \angle C}{c}.$$

Proof. The area of triangle $ABC$ can be written in three different ways: $$[\triangle ABC] = \frac {1}{2} bc \sin \angle A = \frac {1}{2} ac \sin \angle B = \frac {1}{2} ab \sin \angle C,$$ giving us the result. $\blacksquare$

Review of The Law of Sines

The Law of Sines, also known as the Sine Rule, is useful for solving angles of a triangle. The law is as follows: $$\frac{A}{\sin(a)} = \frac{B}{\sin(b)} = \frac{C}{\sin(c)}$$ Where $a,b$ and $c$ are angles, and $A,B$ and $C$ are sides.

What this law states is that when we divide side $a$ by the sine of angle $A$ it is equal to side $b$ by the sine of angle $B$, and also equal to side $c$ divided by the sine of angle $C$

Example of The Law of Sines

To start out this problem, let's write out the law of sines. $$\frac{A}{\sin(a)} = \frac{B}{\sin(b)} = \frac{C}{\sin(c)}$$ We can see from above that the $a, A$ and $C$ are what we will be solving for, so that means we do have our $b, B$ and $c$ terms.

So, plugging these into the law of sines, we will have: $$\frac{A}{\sin(a)} = \frac{7}{\sin(43.2^{\circ})} = \frac{C}{\sin(87.9^{\circ})}$$

We notice that we have no information for neither $A$ nor $a$. However, we can easily find $a$ given simple geometry! We know that the angles of a triangle add up to $180^{\circ}$. If you need a refresher on the addition of angles in a triangle, you can click here! So, we can do $$a = 180 - b - c$$ $$a = 180 - 43.2 - 87.9$$ This will give us an $a$ of $48.9^{\circ}$! Let's update our law of sines: $$\frac{A}{\sin(48.9^{\circ})} = \frac{7}{\sin(43.2^{\circ})} = \frac{C}{\sin(87.9^{\circ})}$$

Now, let's compare our $B/b$ and $C/c$ terms. This will look like: $$\frac{7}{\sin(43.2^{\circ})} = \frac{C}{\sin(87.9^{\circ})}$$

Let's multiply both sides by $\sin(43.2^{\circ}) \sin(87.9^{\circ})$: $$\frac{7}{\sin(43.2^{\circ})} \times \textcolor{red}{\sin(43.2^{\circ}) \sin(87.9^{\circ})} = \frac{C}{\sin(87.9^{\circ})} \times \textcolor{red}{\sin(43.2^{\circ}) \sin(87.9^{\circ})}$$

This will give us: $$7 \sin(87.9^{\circ}) = C \sin(43.2^{\circ})$$

Now, if we divide both sides by $\sin(43.2^{\circ})$, we will find our $C$ term!

$$\frac{7 \sin(87.9^{\circ})}{\textcolor{red}{\sin(43.2^{\circ})}} =\frac{C \sin(43.2^{\circ})}{\textcolor{red}{\sin(43.2^{\circ})}}$$ $$C = \frac{7 \sin(87.9^{\circ})}{\sin(43.2^{\circ})}$$ $$C = 10.2$$

Great! Now we only have $A$ to find. Since we were given the $B/b$ terms in the problem, we know that those are for certain the answer. Let's use them again to find $A$. This will look like:

$$\frac{A}{\sin(48.9^{\circ})} = \frac{7}{\sin(43.2^{\circ})}$$

Let's multiply both sides by $\sin(48.9^{\circ}) \sin(43.2^{\circ})$:

$$\frac{A}{\sin(48.9^{\circ})} \times \textcolor{red}{\sin(48.9^{\circ}) \sin(43.2^{\circ})} = \frac{7}{\sin(43.2^{\circ})} \times \textcolor{red}{\sin(48.9^{\circ}) \sin(43.2^{\circ})}$$

This will give us: $$A \sin(43.2^{\circ}) = 7 \sin(48.9^{\circ})$$

Now, if we divide both sides by $\sin(43.2^{\circ})$, we will find our $A$ term!

$$\frac{A \sin(43.2^{\circ})}{\textcolor{red}{\sin(43.2^{\circ})}} =\frac{7 \sin(48.9^{\circ})}{\textcolor{red}{\sin(43.2^{\circ})}}$$ $$A = \frac{7 \sin(48.9^{\circ})}{\sin(43.2^{\circ})}$$ $$A = 7.7$$

Great! We solved for $a, A$ and $C$! Let's check our ratios, for they should all equal each other!

A: $\frac{7.7}{\sin(48.9^{\circ})} = 10.22$

B: $\frac{7}{\sin(43.2^{\circ})} = 10.22$

C: $\frac{10.2}{\sin(87.9^{\circ})} = 10.21$

Note: $C$ is off by $0.01$, which is not enough to matter for this problem!