# projection formula

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 (http://planetmath.org/ProjectionOfPoint) of a line segment, the truth of the theorem is apparent; the below 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

 $\frac{a}{\cos\alpha}+\frac{b}{\cos\beta}=\frac{c}{\cos\alpha\,\cos\beta}.$
Title projection formula ProjectionFormula 2013-03-22 18:27:11 2013-03-22 18:27:11 pahio (2872) pahio (2872) 9 pahio (2872) Theorem msc 51N99 projection formula for triangles BaseAndHeightOfTriangle