Give the following in polar form:
(i) `sin120^(@)-icos120^(@)`
(ii) `[2cos210^(@)+isin210^(@))][4(cos120^(@)+isin120^(@))]`
(iii) `[3(cos225^(@)+isin225^(@))][6(cos45^(@)+isin45^(@))]`.

Answer

To convert the given expressions into polar form, we will follow the steps outlined in the video transcript. Let's go through each part step by step.

### (i) Convert `sin(120°) - i cos(120°)` into polar form

1. **Rewrite the expression**:
We start with the expression:
\[ \sin(120°) - i \cos(120°) \]
We can use the identity for sine and cosine:
\[ \sin(120°) = \cos(90° - 120°) = \cos(30°) \]
\[ \cos(120°) = -\sin(90° - 120°) = -\sin(30°) \]
Thus, we can rewrite the expression as:
\[ \sin(120°) - i \cos(120°) = \cos(30°) + i \sin(30°) \]

2. **Identify the modulus and argument**:
The modulus \( r \) is:
\[ r = \sqrt{(\cos(30°))^2 + (\sin(30°))^2} = 1 \]
The argument \( \theta \) is:
\[ \theta = 30° \]

3. **Write in polar form**:
Therefore, the polar form is:
\[ 1 \cdot (\cos(30°) + i \sin(30°)) = e^{i \cdot 30°} \]

### (ii) Convert `[2cos(210°) + i sin(210°)][4(cos(120°) + i sin(120°)]` into polar form

1. **Convert each part to polar form**:
The first part:
\[ 2 \cos(210°) + i \sin(210°) = 2 e^{i \cdot 210°} \]
The second part:
\[ 4 (\cos(120°) + i \sin(120°)) = 4 e^{i \cdot 120°} \]

2. **Multiply the two expressions**:
The product is:
\[ (2 e^{i \cdot 210°})(4 e^{i \cdot 120°}) = 8 e^{i (210° + 120°)} = 8 e^{i \cdot 330°} \]

3. **Write in polar form**:
Therefore, the polar form is:
\[ 8 (\cos(330°) + i \sin(330°)) \]

### (iii) Convert `[3(cos(225°) + i sin(225°))][6(cos(45°) + i sin(45°))]` into polar form

1. **Convert each part to polar form**:
The first part:
\[ 3 (\cos(225°) + i \sin(225°)) = 3 e^{i \cdot 225°} \]
The second part:
\[ 6 (\cos(45°) + i \sin(45°)) = 6 e^{i \cdot 45°} \]

2. **Multiply the two expressions**:
The product is:
\[ (3 e^{i \cdot 225°})(6 e^{i \cdot 45°}) = 18 e^{i (225° + 45°)} = 18 e^{i \cdot 270°} \]

3. **Write in polar form**:
Therefore, the polar form is:
\[ 18 (\cos(270°) + i \sin(270°)) \]

### Final Answers:
1. \( e^{i \cdot 30°} \)
2. \( 8 (\cos(330°) + i \sin(330°)) \)
3. \( 18 (\cos(270°) + i \sin(270°)) \)