Convert each of the following complex numbers in polar form:
(i) -3
(ii) `sqrt3+i`
(iii) `i`.

Answer

To convert the given complex numbers into polar form, we will follow these steps for each complex number:

### (i) Convert -3 to polar form

1. **Identify the complex number**: The complex number is -3, which can be written as -3 + 0i.

2. **Calculate the modulus (r)**:
\[ r = |z| = \sqrt{(-3)^2 + 0^2} = \sqrt{9} = 3 \]

3. **Determine the argument (θ)**:
- Since -3 lies on the negative x-axis, the angle θ is:
\[ \theta = \pi \text{ (or 180 degrees)} \]

4. **Write in polar form**:
\[ z = r(\cos \theta + i \sin \theta) = 3(\cos \pi + i \sin \pi) \]

### Polar form of -3:
\[ -3 = 3(\cos \pi + i \sin \pi) \]

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### (ii) Convert \(\sqrt{3} + i\) to polar form

1. **Identify the complex number**: The complex number is \(\sqrt{3} + 1i\).

2. **Calculate the modulus (r)**:
\[ r = |z| = \sqrt{(\sqrt{3})^2 + 1^2} = \sqrt{3 + 1} = \sqrt{4} = 2 \]

3. **Determine the argument (θ)**:
- Using \(\tan \theta = \frac{\text{Imaginary part}}{\text{Real part}} = \frac{1}{\sqrt{3}}\):
\[ \theta = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6} \text{ (or 30 degrees)} \]

4. **Write in polar form**:
\[ z = r(\cos \theta + i \sin \theta) = 2(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}) \]

### Polar form of \(\sqrt{3} + i\):
\[ \sqrt{3} + i = 2(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}) \]

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### (iii) Convert \(i\) to polar form

1. **Identify the complex number**: The complex number is \(0 + 1i\).

2. **Calculate the modulus (r)**:
\[ r = |z| = \sqrt{0^2 + 1^2} = \sqrt{1} = 1 \]

3. **Determine the argument (θ)**:
- Since \(i\) lies on the positive y-axis, the angle θ is:
\[ \theta = \frac{\pi}{2} \text{ (or 90 degrees)} \]

4. **Write in polar form**:
\[ z = r(\cos \theta + i \sin \theta) = 1(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2}) \]

### Polar form of \(i\):
\[ i = 1(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2}) \]

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### Summary of Polar Forms:
1. \(-3 = 3(\cos \pi + i \sin \pi)\)
2. \(\sqrt{3} + i = 2(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6})\)
3. \(i = 1(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2})\)

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