To convert the complex number $z = -\sqrt{3} + i$ into polar form, we need to express it in the form:

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

where $r$ is the magnitude (or modulus) of the complex number and $\theta$ is the argument (or angle).

Step 1: Find the magnitude $r$

The magnitude $r$ is given by: $r = \sqrt{a^2 + b^2}$ where $z = a + bi$. Here, $a = -\sqrt{3}$ and $b = 1$.

$r = \sqrt{(-\sqrt{3})^2 + 1^2} = \sqrt{3 + 1} = \sqrt{4} = 2$

Step 2: Find the argument $\theta$

The argument $\theta$ is found using the formula: $\theta = \tan^{-1}\left(\frac{b}{a}\right)$

However, because $a = -\sqrt{3}$ (negative) and $b = 1$ (positive), the complex number lies in the second quadrant. So, we need to add $\pi$ to the angle found by the arctangent function to get the correct angle in the second quadrant.

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

Simplify $\frac{1}{-\sqrt{3}}$: $\tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) = -\frac{\pi}{6}$ So, $\theta = -\frac{\pi}{6} + \pi = \frac{5\pi}{6}$

Step 3: Write the polar form

Now that we have $r = 2$ and $\theta = \frac{5\pi}{6}$, the polar form of the complex number is:

$z = 2 \left( \cos \frac{5\pi}{6} + i \sin \frac{5\pi}{6} \right)$

So, the polar form of $-\sqrt{3} + i$ is: $\boxed{2 \left( \cos \frac{5\pi}{6} + i \sin \frac{5\pi}{6} \right)}$

Would you like more details or have any other questions?

Here are 5 related questions:

1. How do you convert a complex number from polar form to rectangular form?
2. What is the significance of the magnitude in the polar form of a complex number?
3. How do you add or subtract two complex numbers in polar form?
4. Can you explain the difference between the argument and the principal argument of a complex number?
5. How do you multiply two complex numbers in polar form?

Tip: Always remember to consider the correct quadrant when calculating the argument $\theta$ to ensure the angle is accurate.