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# 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 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

$\frac{a}{\cos\alpha}+\frac{b}{\cos\beta}=\frac{c}{\cos\alpha\,\cos\beta}.$ |

Related:

BaseAndHeightOfTriangle

Synonym:

projection formula for triangles

Type of Math Object:

Theorem

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

51N99*no label found*

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