Change the following complex numbers into polar form
`-4+4 sqrt3i`

Answer

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

### Step 1: Identify the real and imaginary parts
The given complex number is \( z = -4 + 4\sqrt{3}i \).
- Real part (\(x\)) = \(-4\)
- Imaginary part (\(y\)) = \(4\sqrt{3}\)

### Step 2: Calculate the modulus (\(|z|\))
The modulus of a complex number \(z = x + yi\) is given by:
\[ |z| = \sqrt{x^2 + y^2} \]
Substituting the values of \(x\) and \(y\):
\[ |z| = \sqrt{(-4)^2 + (4\sqrt{3})^2} = \sqrt{16 + 48} = \sqrt{64} = 8 \]

### Step 3: Calculate the argument (\(\theta\))
The argument \(\theta\) of a complex number can be found using:
\[ \tan(\theta) = \frac{y}{x} \]
Substituting the values:
\[ \tan(\theta) = \frac{4\sqrt{3}}{-4} = -\sqrt{3} \]
Now, we need to find the angle whose tangent is \(-\sqrt{3}\). The reference angle is:
\[ \alpha = \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \text{ (or } 60^\circ\text{)} \]
Since the real part is negative and the imaginary part is positive, the complex number lies in the second quadrant. Therefore, the argument \(\theta\) is:
\[ \theta = \pi - \alpha = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \]

### Step 4: Write the polar form
The polar form of a complex number is given by:
\[ z = |z| \left( \cos(\theta) + i\sin(\theta) \right) \]
Substituting the values of \(|z|\) and \(\theta\):
\[ z = 8 \left( \cos\left(\frac{2\pi}{3}\right) + i\sin\left(\frac{2\pi}{3}\right) \right) \]

### Final Answer
Thus, the polar form of the complex number \(-4 + 4\sqrt{3}i\) is:
\[ z = 8 \left( \cos\left(\frac{2\pi}{3}\right) + i\sin\left(\frac{2\pi}{3}\right) \right) \]

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