Write 2i in polar form.

Answer

To write the complex number \( 2i \) in polar form, we follow these steps:

### Step 1: Identify the complex number
The given complex number is \( z = 2i \). In standard form, this can be expressed as \( z = 0 + 2i \), where the real part \( x = 0 \) and the imaginary part \( y = 2 \).

### Step 2: Calculate the modulus \( r \)
The modulus \( r \) of a complex number \( z = x + yi \) is given by:
\[ r = \sqrt{x^2 + y^2} \]
Substituting \( x = 0 \) and \( y = 2 \):
\[ r = \sqrt{0^2 + 2^2} = \sqrt{0 + 4} = \sqrt{4} = 2 \]

### Step 3: Calculate the argument \( \theta \)
The argument \( \theta \) (or angle) can be found using the formula:
\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]
Since \( x = 0 \) and \( y = 2 \), we have:
\[ \theta = \tan^{-1}\left(\frac{2}{0}\right) \]
This indicates that the angle is \( \frac{\pi}{2} \) radians (or 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 expressed as:
\[ z = r(\cos \theta + i \sin \theta) \]
Substituting \( r = 2 \) and \( \theta = \frac{\pi}{2} \):
\[ z = 2\left(\cos\frac{\pi}{2} + i \sin\frac{\pi}{2}\right) \]

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

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