Convert of the complex number in the polar form: `i`

Explanation

To convert the complex number \( i \) into polar form, we can follow these steps:

### Step 1: Identify the complex number
The complex number given is \( i \), which can be expressed in the standard form \( x + iy \). Here, \( x = 0 \) and \( y = 1 \).

### Step 2: Calculate the modulus
The modulus \( |z| \) of a complex number \( z = x + iy \) is given by the formula:
\[ |z| = \sqrt{x^2 + y^2} \]
Substituting the values:
\[ |z| = \sqrt{0^2 + 1^2} = \sqrt{1} = 1 \]

### Step 3: Calculate the argument
The argument \( \theta \) (or angle) of the complex number can be calculated using:
\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]
In this case, since \( x = 0 \) and \( y = 1 \):
\[ \theta = \tan^{-1}\left(\frac{1}{0}\right) \]
This means the angle is \( \frac{\pi}{2} \) radians (90 degrees), as the point lies on the positive imaginary axis.

### Step 4: Write the polar form
The polar form of a complex number is given by:
\[ z = |z| (\cos \theta + i \sin \theta) \]
Substituting the values we found:
\[ z = 1 \left( \cos\left(\frac{\pi}{2}\right) + i \sin\left(\frac{\pi}{2}\right) \right) \]
Using the known values of cosine and sine:
\[ \cos\left(\frac{\pi}{2}\right) = 0 \quad \text{and} \quad \sin\left(\frac{\pi}{2}\right) = 1 \]
Thus, we have:
\[ z = 1 \left( 0 + i \cdot 1 \right) = i \]

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