Q8: How does the law of cosines work?

Question:

My teacher wants me to find the sides of a triangle, but it's not a right triangle. I know two sides and one of the angles. How can I find the third side without using the Pythagorean Theorem?

Answer:

You're right -- the Pythagorean Theorem doesn't apply here because it's not a right triangle. Instead, we can use the Law of Cosines to find the missing side:

The Law of Cosines is really a generalized version of the Pythagorean Theorem. If it's a right triangle then the angle C is 90 degrees, so its cosine is 0. Plug that into the above equation and the whole extra term disappears.

In this case the angle C is 30 degrees, and a is 3 and b is 4. You'll notice that it doesn't matter which we call a or b.

Of course, if it was a regular right triangle we know that the length of side C would have been 5. In this case, the length is just 2.05 because the angle is a lot smaller.