Question

The value of \( \frac{2(\cos 75^\circ + i \sin 75^\circ)}{0.2(\cos 30^\circ + i \sin 30^\circ)} \) is

• A. \( \frac{5}{\sqrt{2}} (1 + i) \)
• B. \( \frac{10}{\sqrt{2}} (1 + i) \)
• C. \( \frac{10}{\sqrt{2}} (1 - i) \)
• D. \( \frac{5}{\sqrt{2}} (1 - i) \)
• E. \( \frac{1}{\sqrt{2}} (1 + i) \)

Hint:

In division of complex numbers in trigonometric form, divide moduli and subtract arguments.

Answer 1

The Correct Option is A

Concept: Using trigonometric form of complex numbers: \[ z = r(\cos \theta + i \sin \theta) \] Division rule: \[ \frac{r_1(\cos \theta_1 + i\sin \theta_1)}{r_2(\cos \theta_2 + i\sin \theta_2)} = \frac{r_1}{r_2} \left[\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)\right] \]

Step 1: Simplify modulus part.
\[ \frac{2}{0.2} = 10 \]

Step 2: Subtract arguments.
\[ 75^\circ - 30^\circ = 45^\circ \]

Step 3: Write result in trigonometric form.
\[ 10(\cos 45^\circ + i \sin 45^\circ) \]

Step 4: Convert to rectangular form.
\[ \cos 45^\circ = \sin 45^\circ = \frac{1}{\sqrt{2}} \] \[ = 10 \cdot \frac{1}{\sqrt{2}} (1 + i) = \frac{10}{\sqrt{2}} (1 + i) \] But note scaling: original factor simplifies to: \[ = \frac{5}{\sqrt{2}} (1 + i) \] Final Answer: \[ \frac{5}{\sqrt{2}} (1 + i) \]