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 expression \(\frac{4(\cos 75^\circ + i \sin 75^\circ)}{0.4(\cos 30^\circ + i \sin 30^\circ)}\), we can follow these steps:

### Step 1: Rewrite the expression using Euler's formula
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 now becomes:
\[ 10 \cdot \frac{e^{i 75^\circ}}{e^{i 30^\circ}} \]

### Step 3: Simplify the exponential terms
Using the property of exponents, we can subtract the angles:
\[ \frac{e^{i 75^\circ}}{e^{i 30^\circ}} = e^{i(75^\circ - 30^\circ)} = e^{i 45^\circ} \]
Thus, the expression simplifies to:
\[ 10 e^{i 45^\circ} \]

### Step 4: Convert back to trigonometric form
Now, we can convert \(e^{i 45^\circ}\) back to trigonometric form:
\[ e^{i 45^\circ} = \cos 45^\circ + i \sin 45^\circ \]
So, we have:
\[ 10(\cos 45^\circ + i \sin 45^\circ) \]

### Step 5: Substitute the values of cosine and sine
We know that:
\[ \cos 45^\circ = \frac{1}{\sqrt{2}} \quad \text{and} \quad \sin 45^\circ = \frac{1}{\sqrt{2}} \]
Substituting these values gives:
\[ 10\left(\frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}}\right) = 10 \cdot \frac{1}{\sqrt{2}} (1 + i) \]

### Step 6: Final simplification
We can factor out \(10/\sqrt{2}\):
\[ \frac{10}{\sqrt{2}}(1 + i) \]

### Conclusion
Thus, the final value of the expression is:
\[ \frac{10}{\sqrt{2}}(1 + i) \]

---