Theorem. Let $a$ , $b$ , $c$ be the sides of a triangle and $\alpha$ , $\beta$ the angles opposing $a$ , $b$ , respectively. Then one has $$c = a\cos\beta+b\cos\alpha,$$ independently whether the angles are acute, right or obtuse.

Knowing the way to determine the length of the projection of a line segment, the truth of the theorem is apparent; the below diagrams illustrate the cases where $\beta$ is acute and obtuse (cosine of an obtuse angle is negative).

Note. Especially, if neither of $\alpha$ and $\beta$ is right angle, the formula of the theorem may be written $$\frac{a}{\cos\alpha}+\frac{b}{\cos\beta} = \frac{c}{\cos\alpha\,\cos\beta}.$$