Complete the polar form of
–4+4𝑖√3
Write your answers as decimals rounded to the nearest tenth. Express the argument 𝜃 in degrees, with 0°≤𝜃<360°.

To convert the complex number \(-4 + 4i\sqrt{3}\) to its polar form, we follow these steps:

1. **Find the magnitude \(r\)**:
   \[
   r = \sqrt{a^2 + b^2}
   \]
   where \(a = -4\) and \(b = 4\sqrt{3}\).

   \[
   r = \sqrt{(-4)^2 + (4\sqrt{3})^2} = \sqrt{16 + 48} = \sqrt{64} = 8
   \]

2. **Find the argument \(\theta\)**:
   \[
   \theta = \tan^{-1}\left(\frac{b}{a}\right)
   \]
   Since \(a = -4\) and \(b = 4\sqrt{3}\):

   \[
   \theta = \tan^{-1}\left(\frac{4\sqrt{3}}{-4}\right) = \tan^{-1}(-\sqrt{3})
   \]

   The tangent of \(\theta\) is \(-\sqrt{3}\). This value corresponds to an angle of \(-60^\circ\) (or \(300^\circ\)) in the fourth quadrant. However, since our complex number is in the second quadrant (because \(a\) is negative and \(b\) is positive), we need to add \(180^\circ\) to \(-60^\circ\) to find the correct angle:

   \[
   \theta = 180^\circ - 60^\circ = 120^\circ
   \]

   Therefore, \(\theta = 120^\circ\).

3. **Write the polar form**:
   The polar form of a complex number is given by:
   \[
   r (\cos \theta + i \sin \theta)
   \]

   Substituting the values we found:
   \[
   8 \left( \cos 120^\circ + i \sin 120^\circ \right)
   \]

So, the polar form of \(-4 + 4i\sqrt{3}\) is:
\[
8 (\cos 120^\circ + i \sin 120^\circ)
\]

### Final Answer

\[
\boxed{8 \left( \cos 120^\circ + i \sin 120^\circ \right)}
\]

Would you like more details or have any questions about this solution?

Here are 5 questions you might ask:
1. How do you determine which quadrant the complex number is in?
2. What are some properties of polar coordinates?
3. Can you explain the relationship between polar and Cartesian coordinates?
4. How do you convert a polar form back to Cartesian form?
5. What are some applications of complex numbers in polar form?

**Tip:** When converting complex numbers to polar form, always verify the quadrant to ensure the argument \(\theta\) is correctly calculated.

Learn how to convert the complex number -4 + 4i√3 to its polar form with detailed step-by-step instructions. Understand the calculation of magnitude and argument in degrees. Suitable for advanced high school students.

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