Express in polar form (mod-Amplitude form)
`-1 +i`

Answer

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

### Step 1: Identify the complex number
The given complex number is:
\[ z = -1 + i \]

### Step 2: Calculate the modulus (r)
The modulus \(r\) of a complex number \(z = x + iy\) is given by:
\[ r = \sqrt{x^2 + y^2} \]
For our complex number, \(x = -1\) and \(y = 1\):
\[ r = \sqrt{(-1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Step 3: Calculate the angle (θ)
To find the angle \(\theta\), we use the formulas:
\[ \cos \theta = \frac{x}{r} \quad \text{and} \quad \sin \theta = \frac{y}{r} \]
Substituting the values we have:
\[ \cos \theta = \frac{-1}{\sqrt{2}} \quad \text{and} \quad \sin \theta = \frac{1}{\sqrt{2}} \]

### Step 4: Determine the quadrant
Since \(\cos \theta\) is negative and \(\sin \theta\) is positive, the complex number lies in the second quadrant.

### Step 5: Find the angle θ
In the second quadrant, the reference angle whose cosine is \(-\frac{1}{\sqrt{2}}\) and sine is \(\frac{1}{\sqrt{2}}\) is \(45^\circ\). Therefore, the angle \(\theta\) in the second quadrant is:
\[ \theta = 180^\circ - 45^\circ = 135^\circ \]

### Step 6: Write the polar form
Now we can express the complex number in polar form:
\[ z = r(\cos \theta + i \sin \theta) \]
Substituting \(r\) and \(\theta\):
\[ z = \sqrt{2} \left( \cos 135^\circ + i \sin 135^\circ \right) \]

### Final Result
Thus, the polar form of the complex number \(-1 + i\) is:
\[ z = \sqrt{2} \left( \cos 135^\circ + i \sin 135^\circ \right) \]

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