The law sines states for any triangle with sides A,B, or C that

sin(a)/A = sin(b)/B = sin(c)/C

or equivalently

A/sin(a) = B/sin(b) = C/sin(c)

So if we have a triangle as such, with side x unknown we can find it by using the law of sines:

**Proof of the Law of Sines**

Consider the following diagram:

In the first step of the proof we divide the triangle into two right triangles by drawing a line of length h1.

Now we have two right triangles we can say that

sin(b)=h/A or h=A*sin(b)

similarly

sin(a)=h/B so h=B*sin(a)

now we can see that

B*sin(a)=h=A*sin(b)

divide both sides by AB and we get

sin(a)/A = sin(b)/B which is the first half of the law of sines.

For the next part, we draw a right triangle out from side B, creating a new length h2.

From this new triangle we see that

sin(b)=h/C and h=sin(b)*C

we also see that

sin(180-c) = h/B or h=sin(180-c)*B

but from the unit circle below we see that sin(180-c)=sin(c)

so we can write sin(c)*B=h and from this conclude

sin(b)*C=h=sin(c)*B

again we divide both sides by BC and get

sin(b)/B = sin(c)/C

so we have sin(a)/A=sin(b)/B=sin(c)/C which is the law of sines.

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