The given complex number is $3 + 3\sqrt{3}i$.

Step 1: Convert to Polar Form

A complex number in polar form is expressed as: $z = r \text{cis} \theta = r (\cos \theta + i \sin \theta),$ where:

• $r$ is the magnitude of the complex number.
• $\theta$ is the argument of the complex number, measured in radians.

Magnitude ($r$):

The magnitude is calculated as: $r = \sqrt{x^2 + y^2},$ where $x$ is the real part, and $y$ is the imaginary part.

For $z = 3 + 3\sqrt{3}i$: $r = \sqrt{3^2 + (3\sqrt{3})^2} = \sqrt{9 + 27} = \sqrt{36} = 6.$

Argument ($\theta$):

The argument is: $\theta = \tan^{-1}\left(\frac{y}{x}\right),$ where $y = 3\sqrt{3}$ and $x = 3$.

$\theta = \tan^{-1}\left(\frac{3\sqrt{3}}{3}\right) = \tan^{-1}(\sqrt{3}).$

The value of $\tan^{-1}(\sqrt{3})$ corresponds to $\frac{\pi}{3}$ radians.

Since both $x$ and $y$ are positive, the complex number lies in the first quadrant, so the argument remains $\frac{\pi}{3}$.

Step 2: Write in Polar Form

The polar form of the complex number is: $z = r (\cos \theta + i \sin \theta),$ or simply: $z = 6 \text{cis} \frac{\pi}{3}.$

Final Answer:

The polar form of $3 + 3\sqrt{3}i$ is: $6 \text{cis} \frac{\pi}{3} = 6 (\cos \frac{\pi}{3} + i \sin \frac{\pi}{3}).$

Would you like further clarification or additional examples?

Related Questions:

1. How is the argument $\theta$ adjusted for different quadrants?
2. What are the key differences between rectangular and polar forms?
3. Can polar form be used for multiplication and division of complex numbers?
4. How do we convert polar form back to rectangular form?
5. Why is the magnitude $r$ always a positive value?

Tip:

When working with complex numbers, always ensure the argument $\theta$ is adjusted to the correct quadrant based on the signs of the real and imaginary parts.