The value of `[sqrt(2)(cos(56^(@)15^('))+isin(56^(@)15^('))]^(8)`, is

Options

A. `4i`
B. `8i`
C. `16i`
D. `-16i`

Answer

To solve the problem, we need to evaluate the expression \([ \sqrt{2} (\cos(56^\circ 15') + i \sin(56^\circ 15')) ]^8\).

### Step-by-Step Solution:

1. **Convert Minutes to Degrees**:
- We know that \(1' = \frac{1}{60}^\circ\).
- Therefore, \(15' = \frac{15}{60}^\circ = \frac{1}{4}^\circ = 0.25^\circ\).
- Thus, \(56^\circ 15' = 56^\circ + 0.25^\circ = 56.25^\circ\).

2. **Rewrite the Expression**:
- The expression can be rewritten as:
\[ [\sqrt{2} (\cos(56.25^\circ) + i \sin(56.25^\circ)]^8 \]
- This can be expressed in polar form as:
\[ [\sqrt{2}]^8 \cdot [\cos(56.25^\circ) + i \sin(56.25^\circ)]^8 \]

3. **Calculate \([\sqrt{2}]^8\)**:
- We know that \((\sqrt{2})^8 = (2^{1/2})^8 = 2^{4} = 16\).

4. **Use De Moivre's Theorem**:
- According to De Moivre's theorem, \((\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta)\).
- Thus, we have:
\[ [\cos(56.25^\circ) + i \sin(56.25^\circ)]^8 = \cos(8 \times 56.25^\circ) + i \sin(8 \times 56.25^\circ) \]
- Calculate \(8 \times 56.25^\circ = 450^\circ\).

5. **Evaluate \(\cos(450^\circ)\) and \(\sin(450^\circ)\)**:
- We know that \(450^\circ\) can be simplified:
\[ 450^\circ = 360^\circ + 90^\circ = 90^\circ \]
- Therefore:
\[ \cos(450^\circ) = \cos(90^\circ) = 0 \]
\[ \sin(450^\circ) = \sin(90^\circ) = 1 \]

6. **Combine the Results**:
- Now substituting back, we have:
\[ 16 \cdot [\cos(450^\circ) + i \sin(450^\circ)] = 16 \cdot [0 + i \cdot 1] = 16i \]

7. **Final Answer**:
- The value of the given expression is:
\[ 16i \]