The value of `( 4( cos 75^(@) + isin75^(@)))/(0.4(cos30^(@) + i sin 30^(@)))` is

Options

A. `( sqrt( 2))/( 10) ( 1+i)`
B. `( sqrt( 2))/( 10) ( 1-i)`
C. `(10)/( sqrt(2)) ( 1-i)`
D. `(10)/( sqrt(2)) ( 1+i)`

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: Rewrite the expression using Euler's formula

Using Euler's formula, we can rewrite the expression as:

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

### Step 2: Simplify the coefficients

Next, we simplify the coefficients:

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

So, we can rewrite the expression as:

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

### Step 3: Apply the property of exponents

Using the property of exponents, we can simplify the fraction:

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

### Step 4: Combine the results

Now, we can combine the results:

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

### Step 5: Rewrite in trigonometric form

Using Euler's formula again, we can express \(e^{i 45^\circ}\) in trigonometric form:

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

Thus, we have:

\[ 10 \cdot e^{i 45^\circ} = 10 \left( \frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}} \right) = \frac{10}{\sqrt{2}} + i \frac{10}{\sqrt{2}} \]

### Final Result

So, the final value of the expression is:

\[ \frac{10}{\sqrt{2}} + i \frac{10}{\sqrt{2}} \]