Convert of the complex number in the polar form: `-1 + i`

Explanation

To convert the complex number \(-1 + i\) into polar form, we will follow these steps:

### Step 1: Identify the components
The complex number can be expressed in the form \(x + iy\), where:
- \(x = -1\)
- \(y = 1\)

### Step 2: Calculate the modulus \(r\)
The modulus \(r\) is given by the formula:
\[ r = \sqrt{x^2 + y^2} \]
Substituting the values of \(x\) and \(y\):
\[ r = \sqrt{(-1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Step 3: Calculate the argument \(\theta\)
The argument \(\theta\) can be found using the formula:
\[ \tan \theta = \frac{y}{x} \]
Substituting the values:
\[ \tan \theta = \frac{1}{-1} = -1 \]
The angle whose tangent is \(-1\) is \(-\frac{\pi}{4}\). However, since the point \((-1, 1)\) lies in the second quadrant, we need to adjust the angle:
\[ \theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \]

### Step 4: Write the polar form
Now that we have \(r\) and \(\theta\), we can express the complex number in polar form:
\[ z = r(\cos \theta + i \sin \theta) \]
Substituting the values we found:
\[ z = \sqrt{2} \left( \cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4} \right) \]

### Final Result
Thus, the polar form of the complex number \(-1 + i\) is:
\[ z = \sqrt{2} \left( \cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4} \right) \]

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