Final answer:
To solve 5[cos(π/4) + i sin (π/4)]^3, we utilize De Moivre's Theorem to get (-√2/2 + i√2/2)^3. After evaluation, we adjust for the coefficient by multiplying by 5^3, leading to the final answer of -62.5 - 62.5i.
Step-by-step explanation:
The expression given is 5[cos(π/4) + i sin (π/4)] raised to the 3rd power. To solve this, we can use De Moivre's Theorem which states that (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ). For our case, θ = π/4 and n = 3, so we can evaluate the expression as follows:
- Multiply the angle by 3: π/4 * 3 = 3π/4.
- Find the cosine and sine of 3π/4: cos(3π/4) = -√2/2 and sin(3π/4) = √2/2.
- Apply the theorem: (cos(3π/4) + i sin(3π/4))^3 = (-√2/2 + i√2/2)^3.
- Since the original expression has a coefficient of 5, we multiply the result by 5^3.
Therefore, the final evaluated expression is -125(1/2 + i/2), which simplifies to -62.5 - 62.5i.