# 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

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

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

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

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 |