[Math] Pythagoras’s theorem as a special case of the law of cosines

Question

I heard that the Pythagorean theorem is a special case of the more general theorem relating the lengths of sides in any triangle, the law of cosines?

Answer 1

The law of cosines is: $$c^2 = a^2 + b^2 \;-\; 2\!\cdot\!a\!\cdot\!b\!\cdot\!\cos\theta$$ where $\theta$ is the angle between the sides $a$ and $b$.
Now, when this angle is a right angle ($90^\circ$, or $\frac{\pi}{2}$), its cosine is $0$, so the entire last term is multiplied by $0$ and vanishes, leaving only the usual Pythagorean Theorem for the right triangle with legs $a$ and $b$ and the hypotenuse $c$: $$c^2 = a^2 + b^2$$ Simple as that.