The value of `(4(cos75^(@) + isin 75^(@)))/(0.4(cos30^(@) + i sin 30^(@)))` is :

Options

A. `10/sqrt(20 ( 1 + i)`
B. `10/sqrt(2) ( 1- i)`
C. `5/sqrt(2) ( 1 _ i)`
D. None of these

Answer

To solve the problem, we need to simplify the expression:

\[ \frac{4(\cos 75^\circ + i \sin 75^\circ)}{0.4(\cos 30^\circ + i \sin 30^\circ)} \]

### Step 1: Convert to exponential form
Using Euler's formula, we can express the trigonometric functions in terms of exponentials:

\[ \cos \theta + i \sin \theta = e^{i\theta} \]

Thus, we can rewrite the expression as:

\[ \frac{4 e^{i 75^\circ}}{0.4 e^{i 30^\circ}} \]

### Step 2: Simplify the coefficients
We can simplify the coefficients in the expression:

\[ \frac{4}{0.4} = \frac{4 \times 10}{4} = 10 \]

So the expression becomes:

\[ 10 \cdot \frac{e^{i 75^\circ}}{e^{i 30^\circ}} \]

### Step 3: Use properties of exponents
Using the property of exponents \(\frac{e^a}{e^b} = e^{a-b}\):

\[ \frac{e^{i 75^\circ}}{e^{i 30^\circ}} = e^{i(75^\circ - 30^\circ)} = e^{i 45^\circ} \]

### Step 4: Combine the results
Now we can combine the results:

\[ 10 \cdot e^{i 45^\circ} \]

### Step 5: Convert back to trigonometric form
Using Euler's formula again:

\[ e^{i 45^\circ} = \cos 45^\circ + i \sin 45^\circ \]

We know that \(\cos 45^\circ = \sin 45^\circ = \frac{1}{\sqrt{2}}\), so:

\[ e^{i 45^\circ} = \frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}} \]

### Step 6: Multiply by 10
Now we multiply by 10:

\[ 10 \left( \frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}} \right) = \frac{10}{\sqrt{2}} + i \frac{10}{\sqrt{2}} \]

### Final Result
Thus, the final answer is:

\[ \frac{10}{\sqrt{2}} (1 + i) \]