write polar form of this complex number
`-1+i`

Explanation

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

### Step 1: Identify the complex number
We have the complex number:
\[ z = -1 + i \]

### Step 2: Determine the modulus \(r\)
The modulus \(r\) of a complex number \(z = a + bi\) is given by:
\[ r = |z| = \sqrt{a^2 + b^2} \]
For our complex number:
- \(a = -1\)
- \(b = 1\)

Calculating \(r\):
\[ r = \sqrt{(-1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Step 3: Determine the argument \(\theta\)
The argument \(\theta\) can be found using the formula:
\[ \tan(\theta) = \frac{b}{a} \]
Substituting the values:
\[ \tan(\theta) = \frac{1}{-1} = -1 \]

Now, we need to find the angle whose tangent is \(-1\). The reference angle \(\alpha\) is:
\[ \alpha = \tan^{-1}(1) = \frac{\pi}{4} \]
Since the complex number \(-1 + i\) is located in the second quadrant (where \(a < 0\) and \(b > 0\)), we adjust \(\theta\) as follows:
\[ \theta = \pi - \alpha = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \]

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

### Step 5: Alternative representation
We can also write this in the alternative form:
\[ z = \sqrt{2} \text{cis} \frac{3\pi}{4} \]
where \(\text{cis} \theta = \cos \theta + i \sin \theta\).

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

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