# The rectangular coordinates of a point are given (2, -2). Find polar coordinates of the point. Express theta in radians.

The rectangular coordinates of a point are given (2, -2). Find polar coordinates of the point. Express theta in radians.
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Let the given rectangular coordinates be (x, y). Then, $\left(x,y\right)=\left(2,-2\right)$ Let polar coordinates be $=\left(r,\theta \right)$ Then using formulas - $r=\sqrt{{x}^{2}+{y}^{2}}$
$\theta ={\mathrm{tan}}^{-1}\left(\frac{y}{x}\right)$ Using equation (1) - $r=\sqrt{{x}^{2}+{y}^{2}}$
$r=\sqrt{\left(2{\right)}^{2}+\left(-2{\right)}^{2}}$
$r=\sqrt{4+4}$
$r=\sqrt{8}$
$r=2\sqrt{2}$ Now $\theta ={\mathrm{tan}}^{-1}\left(\frac{y}{x}\right)$
$\theta ={\mathrm{tan}}^{-1}\left(\frac{-2}{2}\right)$
$\theta ={\mathrm{tan}}^{-1}\left(-1\right)$
$\theta ={\mathrm{tan}}^{-1}\left(1\right)\left[{\mathrm{tan}}^{-1}\left(-x\right)=-{\mathrm{tan}}^{-1}\left(x\right)\right]$
$\theta ={\mathrm{tan}}^{-1}\left(\mathrm{tan}\pi /4\right)$
$\theta =-\pi /4$ Hence, Polar coordinates = $\left(r,\theta \right)=\left(2\sqrt{2},-\frac{\pi }{4}\right)$