(ii) Write polar form of 3i.

Answer

To write the polar form of the complex number \(3i\), we will follow these steps:

### Step 1: Identify the components of the complex number
The complex number can be expressed in the form \(z = a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. For \(3i\), we have:
- \(a = 0\)
- \(b = 3\)

### Step 2: Calculate the modulus \(r\)
The modulus \(r\) of a complex number is given by the formula:
\[ r = \sqrt{a^2 + b^2} \]
Substituting the values of \(a\) and \(b\):
\[ r = \sqrt{0^2 + 3^2} = \sqrt{0 + 9} = \sqrt{9} = 3 \]

### Step 3: Calculate the argument \(\theta\)
The argument \(\theta\) of a complex number can be found using the relationships:
\[ \cos \theta = \frac{a}{r}, \quad \sin \theta = \frac{b}{r} \]
Substituting the values we have:
\[ \cos \theta = \frac{0}{3} = 0, \quad \sin \theta = \frac{3}{3} = 1 \]
From \(\cos \theta = 0\), we know that \(\theta\) can be \(\frac{\pi}{2}\) (90 degrees) since cosine is zero at this angle.

### 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 \(\theta\):
\[ z = 3\left(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2}\right) \]

### Final Answer
Thus, the polar form of \(3i\) is:
\[ z = 3\left(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2}\right) \]

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