Find-0-lt-lt-2pi-with-x-y-R-x-sin-y-cos-

Question Number 45941 by arcana last updated on 19/Oct/18

Find 0 < θ < 2 π with x, y \in R

x \cdot s i n θ = y \cdot c o s θ

Answered by MJS last updated on 19/Oct/18

for x = 0 \lor y = 0 {it}^{'} s trivial

x \neq 0 \land y \neq 0 for the following

\sin θ = \frac{y}{x} \cos θ

\tan θ = \frac{y}{x}

generally θ = z π + \arctan \frac{y}{x}; z \in Z

but we need θ \in [0, 2 π [

- \frac{π}{2} < \arctan \frac{y}{x} < \frac{π}{2}

so we have

\frac{y}{x} < 0 \Rightarrow θ = π + \arctan \frac{y}{x} \lor θ = 2 π + \arctan \frac{y}{x}

\frac{y}{x} > 0 \Rightarrow θ = \arctan \frac{y}{x} \lor θ = π + \arctan \frac{y}{x}

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