Question

Evaluate the following expression: \[ \left[\sqrt{2} \left( \cos 56^\circ 15' + i \sin 56^\circ 15' \right)\right]^8 \]

• A. \( 1 \)
• B. \( i \)
• C. \( 16 \)
• D. \( 16i \)

Hint:

When dealing with powers of complex numbers in polar form, use De Moivre's Theorem to simplify the process.

Answer 1

The Correct Option is D

We are given a complex number in polar form \( \sqrt{2} \left( \cos 56^\circ 15' + i \sin 56^\circ 15' \right) \). We need to compute its 8th power.
Step 1: Apply De Moivre's Theorem.
De Moivre's Theorem states that for a complex number \( r(\cos \theta + i \sin \theta) \), its \( n \)-th power is: \[ \left[ r(\cos \theta + i \sin \theta) \right]^n = r^n (\cos(n\theta) + i \sin(n\theta)) \] In our case, \( r = \sqrt{2} \) and \( \theta = 56^\circ 15' \). We need to calculate: \[ \left[\sqrt{2} \left( \cos 56^\circ 15' + i \sin 56^\circ 15' \right)\right]^8 = (\sqrt{2})^8 \left( \cos(8 \times 56^\circ 15') + i \sin(8 \times 56^\circ 15') \right) \]
Step 2: Simplify the expression. First, simplify \( (\sqrt{2})^8 \): \[ (\sqrt{2})^8 = 2^4 = 16 \] Next, calculate \( 8 \times 56^\circ 15' \): \[ 8 \times 56^\circ 15' = 450^\circ \] Thus, the expression becomes: \[ 16 \left( \cos 450^\circ + i \sin 450^\circ \right) \] Since \( 450^\circ = 360^\circ + 90^\circ \), we have: \[ \cos 450^\circ = \cos 90^\circ = 0, \quad \sin 450^\circ = \sin 90^\circ = 1 \] So, the expression simplifies to: \[ 16 \left( 0 + i \right) = 16i \]
Step 3: Conclusion.
Thus, the value of the expression is \( 16i \), and the correct answer is option (4).