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.

Learn how to convert the complex number -sqrt(3) + i into polar form step-by-step. Understand the concepts of magnitude and argument of complex numbers. Suitable for high school students studying complex numbers.

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