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