Ayaan Barr

2022-07-09

What is the polar form of $\left(-2,3\right)$ ?

Charlize Manning

Expert

Step 1
To solve 1. we use Pythagoras Theorem
$r=\sqrt{{\left(-2\right)}^{2}+{3}^{2}}$
$=\sqrt{13}$
To solve 2. we first find the quadrant that the point lies in.
y is positive while x is negative $⇒$ quadrant II
Then we find the basic angle by taking inverse tangent of $|\frac{y}{x}|$
$\alpha ={\mathrm{tan}}^{-1}\left(|\frac{3}{-2}|\right)$
$={\mathrm{tan}}^{-1}\left(\frac{3}{2}\right)$
The angle that we are looking for would be
$\theta =\pi -\alpha$
$=\pi -{\mathrm{tan}}^{-1}\left(\frac{3}{2}\right)$
$\approx 2.16$
Therefore, the polar coordinate is $\left(\sqrt{13},\pi -{\mathrm{tan}}^{-1}\left(\frac{3}{2}\right)\right)$
Note that the answer above is not unique. You can add any integer multiples of $2\pi$ to $\theta$ to get other representations of the same point.

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