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

$$\mathrm{Find}0<\theta <2\pi \mathrm{with}x,y\in \mathbb{R}$$ $$$$ $$x\cdot sin\theta =y\cdot cos\theta$$

Answered by MJS last updated on 19/Oct/18

$$\mathrm{for}x=0\vee y=0{\mathrm{it}}^{\prime }\mathrm{s}\mathrm{trivial}$$ $$x\ne 0\wedge y\ne 0\mathrm{for}\mathrm{the}\mathrm{following}$$ $$\sin \theta =\frac{y}{x}\cos \theta$$ $$\tan \theta =\frac{y}{x}$$ $$\mathrm{generally}\theta =z\pi +\arctan \frac{y}{x};z\in \mathbb{Z}$$ $$\mathrm{but}\mathrm{we}\mathrm{need}\theta \in [0,2\pi [$$ $$-\frac{\pi }{2}<\arctan \frac{y}{x}<\frac{\pi }{2}$$ $$\mathrm{so}\mathrm{we}\mathrm{have}$$ $$\frac{y}{x}<0\Rightarrow \theta =\pi +\arctan \frac{y}{x}\vee \theta =2\pi +\arctan \frac{y}{x}$$ $$\frac{y}{x}>0\Rightarrow \theta =\arctan \frac{y}{x}\vee \theta =\pi +\arctan \frac{y}{x}$$

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