How do I find the angles of a triangle if I only have the lengths of the sides?

Question

Is it possible to find the angles of a triangle if I only have its sides? If so, how can I achieve this?

Regarding my knowledge of triangles: Whilst I was taught trigonometry a few years ago, I cannot for the life of me remember how to do things like use SOHCAHTOA to figure out the length of a side given an angle and a side. I know it's possible and if that were my problem I would continue searching the internet for a solution, but I gather finding an angle without knowing any of the angles is more difficult.

Any input is appreciated, and I thank you for your time in advance.

Answer 1

Use the Cosine Law.

Let $\triangle ABC$ have sides $a$, $b$, and $c$. We are using the usual convention that the length of the side opposite vertex $A$ is called $a$, and so on.

Let $\theta=\angle C$. Then the Cosine Law says that $$c^2=a^2+b^2-2ab\cos \theta.$$ Since we know $a$, $b$, and $c$, we can use the above formula to calculate $\cos\theta$. Then we can use the $\cos^{-1}$ button on the calculator to find $\theta$ to excellent accuracy.

We can use the Cosine Law three times to get the three angles. But we only need to do the calculation for two of the angles: If we have them, the third can be easily found.