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We consider the Complex numbers x,y, z with modulus 1 with the property that: x + y + z = 0 and the set A = {(a, 6,y) E R, sin a + sin 3+ sin y = 0 and cos a + cos 8+ cos y = 0} %3D a) SHow that: x2 + y + z = 0 xy + yz + zx = b) For (a, B, y) E A, calculate: sin 2a + sin 2B+ sin 2y

We consider the Complex numbers x,y, z with modulus 1 with the property that: x + y + z = 0 and the set A = {(a, 6,y) E R, sin a + sin 3+ sin y = 0 and cos a + cos 8+ cos y = 0} %3D a) SHow that: x2 + y + z = 0 xy + yz + zx = b) For (a, B, y) E A, calculate: sin 2a + sin 2B+ sin 2y

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section: Chapter Questions
Problem 22RE
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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.

Please check the attached picture for details.

Please write the complete solution.

Thanks in advance.

We consider the Complex numbers x,y, z with modulus 1 with the property that: x + y + z = 0 and the set A = {(a, B, 7) E R°, sin a + sin B + sin y = 0 and cos a + cos B+ cos y = 0} a) SHow that: x2 + y + z = xy+ yz + za = 0 b) For (a, B, y) E A, calculate: sin 2a + sin 2B+ sin 2y c) SHow that if n E N* then, for every (a, B, 7) E A, sin na + sin nß+ sin ny O if and only if n " is not a multiple of 3.
Transcribed Image Text:We consider the Complex numbers x,y, z with modulus 1 with the property that: x + y + z = 0 and the set A = {(a, B, 7) E R°, sin a + sin B + sin y = 0 and cos a + cos B+ cos y = 0} a) SHow that: x2 + y + z = xy+ yz + za = 0 b) For (a, B, y) E A, calculate: sin 2a + sin 2B+ sin 2y c) SHow that if n E N* then, for every (a, B, 7) E A, sin na + sin nß+ sin ny O if and only if n " is not a multiple of 3.
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