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
Canonical name	ProjectionFormula
Date of creation	2013-03-22 18:27:11
Last modified on	2013-03-22 18:27:11
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	9
Author	pahio (2872)
Entry type	Theorem
Classification	msc 51N99
Synonym	projection formula for triangles
Related topic	BaseAndHeightOfTriangle