Why does $b^2 = c^2 + a^2 - 2ca\cos(B)$ in trigonometry?

Question

https://i.stack.imgur.com/l0Dw7.png

I have a (what I believe to be an isosceles) triangle and the formula $b^2 = c^2 + a^2 - 2ca \cos(B)$ and I just have to "prove it".

Now this really confused me as I'm not used to working with trig without knowing at least two of the values and I'm struggling to find anything about it online. I would appreciate it if somebody could explain how it works, Thanks.

When dealing with diagrams, its best not to trust your eyes :) There is no reason to believe it is isosceles, and fortunately no need for it to be isosceles. — rschwieb, Feb 06 '13 at 14:04

score 4 · Accepted Answer · edited Apr 13 '17 at 12:20

4

The formula is the Law of Cosines, and it works in any Euclidean triangle.

I'm sure you will find lots of information by searching for "proof of law of cosines" or somesuch.

In fact, there are bound to be a few helpful posts at our site. Here is one I found:

What is the most elegant and simple proof for the law of cosines?

edited Apr 13 '17 at 12:20

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answered Feb 06 '13 at 14:01

rschwieb

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Why does $b^2 = c^2 + a^2 - 2ca\cos(B)$ in trigonometry?

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