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

Yup: it states that $$ a=\sqrt{b^2+c^2-2bc\cos\alpha} $$ in standard trigonometric notation (where $a,b,c$ are the sides of the triangle, and $\alpha$ is the angle which is opposite to $a$).

A simple proof can be given with vector calculus: being $\vec{a}=\vec{b}-\vec{c}$, squaring this relation you obtain: $$ \vec{a}\cdot\vec{a}\equiv a^2 = b^2 + c^2 - 2 \vec{b}\cdot \vec{c} $$ which is the formula above substituting the definition of scalar product!