Polar form of $$$81 i$$$

Find the polar form of $$$81 i$$$.

The standard form of the complex number is $$$81 i$$$.

For a complex number $$$a + b i$$$, the polar form is given by $$$r \left(\cos{\left(\theta \right)} + i \sin{\left(\theta \right)}\right)$$$, where $$$r = \sqrt{a^{2} + b^{2}}$$$ and $$$\theta = \operatorname{atan}{\left(\frac{b}{a} \right)}$$$.

We have that $$$a = 0$$$ and $$$b = 81$$$.

Thus, $$$r = \sqrt{0^{2} + 81^{2}} = 81$$$.

Also, $$$\theta = \operatorname{atan}{\left(\frac{81}{0} \right)} = \frac{\pi}{2}$$$.

Therefore, $$$81 i = 81 \left(\cos{\left(\frac{\pi}{2} \right)} + i \sin{\left(\frac{\pi}{2} \right)}\right)$$$.

$$$81 i = 81 \left(\cos{\left(\frac{\pi}{2} \right)} + i \sin{\left(\frac{\pi}{2} \right)}\right) = 81 \left(\cos{\left(90^{\circ} \right)} + i \sin{\left(90^{\circ} \right)}\right)$$$A