Convert the following into polar form :
(i) `-1+isqrt(3)`
(ii) `1-i`
(iii) `1-(1)/(i)`
(iv) `3-4i`
(v) `sin120^(@)-icos120^(@)`
(vi) 2

Explanation

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

1. Identify the real part \( a \) and the imaginary part \( b \) of the complex number \( z = a + bi \).
2. Calculate the modulus \( r \) using the formula \( r = \sqrt{a^2 + b^2} \).
3. Determine the angle \( \theta \) using the formula \( \theta = \tan^{-1}(\frac{b}{a}) \) and adjust the angle based on the quadrant in which the complex number lies.
4. Write the polar form as \( z = r(\cos \theta + i \sin \theta) \).

Now, let's solve each part step by step:

### (i) For the complex number \( -1 + i\sqrt{3} \)

1. Identify \( a = -1 \) and \( b = \sqrt{3} \).
2. Calculate \( r \):
\[ r = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]
3. Calculate \( \theta \):
\[ \tan \theta = \frac{\sqrt{3}}{-1} \implies \theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \]
4. Write the polar form:
\[ z = 2\left(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}\right) \]

### (ii) For the complex number \( 1 - i \)

1. Identify \( a = 1 \) and \( b = -1 \).
2. Calculate \( r \):
\[ r = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]
3. Calculate \( \theta \):
\[ \tan \theta = \frac{-1}{1} \implies \theta = \frac{7\pi}{4} \text{ (fourth quadrant)} \]
4. Write the polar form:
\[ z = \sqrt{2}\left(\cos\frac{7\pi}{4} + i\sin\frac{7\pi}{4}\right) \]

### (iii) For the complex number \( 1 - \frac{1}{i} \)

1. Simplify \( 1 - \frac{1}{i} = 1 + i \) (since \( \frac{1}{i} = -i \)).
2. Identify \( a = 1 \) and \( b = 1 \).
3. Calculate \( r \):
\[ r = \sqrt{1^2 + 1^2} = \sqrt{2} \]
4. Calculate \( \theta \):
\[ \tan \theta = \frac{1}{1} \implies \theta = \frac{\pi}{4} \text{ (first quadrant)} \]
5. Write the polar form:
\[ z = \sqrt{2}\left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right) \]

### (iv) For the complex number \( 3 - 4i \)

1. Identify \( a = 3 \) and \( b = -4 \).
2. Calculate \( r \):
\[ r = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
3. Calculate \( \theta \):
\[ \tan \theta = \frac{-4}{3} \implies \theta = \tan^{-1}\left(-\frac{4}{3}\right) \text{ (fourth quadrant)} \]
4. Write the polar form:
\[ z = 5\left(\cos(\tan^{-1}(-\frac{4}{3})) + i\sin(\tan^{-1}(-\frac{4}{3}))\right) \]

### (v) For the complex number \( \sin 120^\circ - i \cos 120^\circ \)

1. Rewrite using known values:
\[ \sin 120^\circ = \frac{\sqrt{3}}{2}, \quad \cos 120^\circ = -\frac{1}{2} \]
Thus, the complex number becomes:
\[ \frac{\sqrt{3}}{2} + i\left(-\left(-\frac{1}{2}\right)\right) = \frac{\sqrt{3}}{2} + i\frac{1}{2} \]
2. Identify \( a = \frac{\sqrt{3}}{2} \) and \( b = \frac{1}{2} \).
3. Calculate \( r \):
\[ r = \sqrt{\left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{1}{2}\right)^2} = \sqrt{\frac{3}{4} + \frac{1}{4}} = \sqrt{1} = 1 \]
4. Calculate \( \theta \):
\[ \tan \theta = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} \implies \theta = \frac{\pi}{6} \text{ (first quadrant)} \]
5. Write the polar form:
\[ z = 1\left(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\right) \]

### (vi) For the complex number \( 2 \)

1. Identify \( a = 2 \) and \( b = 0 \).
2. Calculate \( r \):
\[ r = \sqrt{2^2 + 0^2} = \sqrt{4} = 2 \]
3. Calculate \( \theta \):
\[ \tan \theta = \frac{0}{2} \implies \theta = 0 \]
4. Write the polar form:
\[ z = 2\left(\cos 0 + i\sin 0\right) \]

### Summary of Polar Forms

1. \( z = 2\left(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}\right) \)
2. \( z = \sqrt{2}\left(\cos\frac{7\pi}{4} + i\sin\frac{7\pi}{4}\right) \)
3. \( z = \sqrt{2}\left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right) \)
4. \( z = 5\left(\cos(\tan^{-1}(-\frac{4}{3})) + i\sin(\tan^{-1}(-\frac{4}{3}))\right) \)
5. \( z = 1\left(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\right) \)
6. \( z = 2\left(\cos 0 + i\sin 0\right) \)