Answered: We consider the Complex numbers x,y, z…

Trigonometry (11th Edition)

11th Edition

ISBN:9780134217437

Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Chapter1: Trigonometric Functions

Section: Chapter Questions

Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.

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Question

We consider the Complex numbers $x, y, z$ with modulus 1 with the property that:
$$
x+y+z=0
$$
and the set $$A=\left\{(\alpha, \beta, \gamma) \in \mathbb{R}^{3}, \sin \alpha+\sin \beta+\sin \gamma=0\right.\hspace{0.2cm}\text{and}\cos \alpha+\cos \beta+\cos \gamma=0\}$$
a) SHow that:
$$x^{2}+y^{2}+z^{2}=x y+y z+z x=0$$
b) For $(\alpha, \beta, \gamma) \in A$, calculate:
$$
\sin 2 \alpha+\sin 2 \beta+\sin 2 \gamma
$$
c) SHow that if $n \in \mathbb{N}^{*}$ then, for every
$$
(\alpha, \beta, \gamma) \in A \text {, }
$$
$\sin n \alpha+\sin n \beta+\sin n \gamma=0$ if and only if " $n$ " is not a multiple of 3.