So we recently covered the Pythagorean Theorem, and I am betting you’ve got at least one concern. No, I don’t mean about whether or not Planet X has begun cutting a destructive swath across our solar system. I mean in regards to calculating the length of the side of a triangle if you know the other two. Don’t have a right triangle? You may think you’re SOL. I am here to tell you there is hope. Enter the Law of Cosines.

**The Law of Cosines**

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“Wait a minute!” you may be thinking. “The *LAW *of cosines?! Why did we bother with Pythagoras and his measly theorem when we had a capital L Law at our disposal?”

Well, while it is true that the Law of Cosines allows us to calculate lengths of sides for non-right triangles, the truth is that is that it is based on the Pythagorean Theorem and not vice versa as some might expect. Let me explain.

## Deriving the Law of Cosines

Consider the following triangle:

This triangle has three sides: a, b, and c. The angles are named respective to the side they are opposite: A, B, and C. We know the lengths of sides a and b, and we know the size of angle C. We want to calculate the length of side c. How can we do it?

Well, it’s not a right triangle so we can’t use the Pythagorean Theorem. Or can we? Let’s drop a line from vertex B down and perpendicular to side b.

Now we’ve got something we can work with! Specifically, we have two right triangles that share a common side, d. Note that I’ve also divided side b into sides b_{1} and b_{2}. With the Pythagorean Theorem, some trig and some algebra we should now be able to solve for c.

Looking the right triangle dcb_{2}, we know by the Pythagorean Theorem that:

That looks like a good start. We’ve got an equation for , and we’ve got two terms, and that we can redefine in terms of our known angle C using the the definitions of sine and cosine on the right triangle adb_{1}.

is easy:

is a little trickier since it is related to through . Specifically, we know that so,

Unfortunately, while we know b (that’s how we defined our problem), we do not know . We can use the definition of cosine, though:

So,

Now that we have and defined in terms of known values, we can substitute both back into our equation for , i.e., the Pythagorean Theorem applied to triangle dcb_{2}.

Hopefully expanding then simplifying the above will get us where we need to go:

There you have it, the Law of Cosines. You’ll notice that the Pythagorean Theorem is a degenerate instance of this. When is 90 degrees, is zero making that whole last term equal to zero, leaving you with . And even though the Pythagorean Theorem is a special case of the Law of Cosines, the Pythagorean Theorem was integral, along with the definitions of sine and cosine, for deriving the more general Law of Cosines.