find-a-b-c-in-R-cos-a-cos-b-cos-c-0-sin-a-sin-b-sin-c-0-

Question Number 59997 by aliesam last updated on 16/May/19

f i n d (a, b, c) i n R

c o s (a) + c o s (b) + c o s (c) = 0

s i n (a) + s i n (b) + s i n (c) = 0

Answered by MJS last updated on 17/May/19

we have only 2 equations in 3 variables \Rightarrow

we are free to choose one

let a = 0

1 + \cos b + \cos c = 0

\sin b + \sin c = 0

let \cos b = x \land \cos c = y \Rightarrow \sin b = \pm \sqrt{1 - x^{2}} \land \sin c = \pm \sqrt{1 - y^{2}}

1 + x + y = 0 \Rightarrow y = - x - 1

\sqrt{1 - x^{2}} - \sqrt{1 - y^{2}} = 0 \Rightarrow 1 - x^{2} = 1 - y^{2}

1 - x^{2} = 1 - {(- x - 1)}^{2}

1 - x^{2} = - 2 x - x^{2}

x = - \frac{1}{2} \Rightarrow y = - \frac{1}{2} \Rightarrow

\Rightarrow \cos b = \cos c = - \frac{1}{2} \Rightarrow

\Rightarrow b = c = \pm \frac{2 π}{3} + 2 π z \land z \in Z but \sin b = - \sin c \Rightarrow

a = α \in R

b = α \pm \frac{2 π}{3} + 2 π z_{1} \land c = α \mp \frac{2 π}{3} + 2 π z_{2} \land z_{1}, z_{2} \in Z

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