To express the complex number $3\sqrt{3} + i$ in polar form, we need to find its modulus and argument. The general polar form of a complex number $z = x + iy$ is:

$z = r (\cos \theta + i \sin \theta)$

Where:

• $r$ is the modulus of the complex number.
• $\theta$ is the argument (angle) of the complex number.

Step 1: Find the modulus $r$

The modulus of a complex number $z = x + iy$ is given by:

$r = \sqrt{x^2 + y^2}$

For $3\sqrt{3} + i$, the real part $x = 3\sqrt{3}$ and the imaginary part $y = 1$.

$r = \sqrt{(3\sqrt{3})^2 + 1^2}$ $r = \sqrt{27 + 1} = \sqrt{28} = 2\sqrt{7}$

Step 2: Find the argument $\theta$

The argument $\theta$ is given by:

$\theta = \tan^{-1}\left( \frac{y}{x} \right)$

For our complex number:

$\theta = \tan^{-1}\left( \frac{1}{3\sqrt{3}} \right)$ $\theta = \tan^{-1}\left( \frac{1}{3\sqrt{3}} \right) = \tan^{-1}\left( \frac{1}{3 \times 1.732} \right) = \tan^{-1}\left( \frac{1}{5.196} \right)$ $\theta = \tan^{-1}(0.192) \approx 10.9^\circ$

Step 3: Express in polar form

Now that we have the modulus and argument, the polar form is:

$z = 2\sqrt{7} \left( \cos 10.9^\circ + i \sin 10.9^\circ \right)$

Alternatively, we can express it as:

$z = 2\sqrt{7} e^{i 10.9^\circ}$

This is the complex number $3\sqrt{3} + i$ in polar form.

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Would you like any further details or clarifications? Here are some related questions that might help:

1. How do you convert a complex number from polar to rectangular form?
2. How do you calculate the modulus of a complex number?
3. What is the geometric interpretation of the argument of a complex number?
4. How do you multiply two complex numbers in polar form?
5. How do you find the argument of a complex number when it lies in different quadrants?

Tip: When calculating the argument, always consider the correct quadrant of the complex number to ensure the angle is accurate.