Question

If $Z=r(\cos\theta+i\sin\theta)$, $(\theta \neq -\pi/2)$ is a solution of $x^3 = i$, then $r^9(\cos(9\theta)+i\sin(9\theta)) =$

• A. $\frac{\sqrt{3}}{2} + \frac{1}{2}i$
• B. 1
• C. $-i$
• D. $-\frac{\sqrt{3}}{2} + \frac{1}{2}i$

Hint:

De Moivre's Theorem, $[r(\cos\theta + i\sin\theta)]^n = r^n(\cos(n\theta) + i\sin(n\theta))$, is fundamental for computing powers and roots of complex numbers. Recognizing its application can greatly simplify complex expressions.

Answer 1

The Correct Option is C

Step 1: Use De Moivre's Theorem to simplify the target expression.
The expression we need to find is $r^9(\cos(9\theta)+i\sin(9\theta))$.
According to De Moivre's Theorem, $(\cos\phi + i\sin\phi)^n = \cos(n\phi) + i\sin(n\phi)$.
So, we can write the expression as:
$r^9(\cos(9\theta)+i\sin(9\theta)) = (r(\cos\theta+i\sin\theta))^9$.

Step 2: Relate the expression to the given variable Z.
We are given that $Z = r(\cos\theta+i\sin\theta)$.
Substituting this into our simplified expression from Step 1, we get:
$(r(\cos\theta+i\sin\theta))^9 = Z^9$.
So, the problem is reduced to finding the value of $Z^9$.

Step 3: Use the given condition that Z is a solution of $x^3 = i$.
Since Z is a solution to the equation $x^3 = i$, it must satisfy the equation.
Therefore, $Z^3 = i$.

Step 4: Calculate $Z^9$ using the result from Step 3.
We can write $Z^9$ as $(Z^3)^3$.
Substituting $Z^3 = i$, we have:
$Z^9 = (i)^3 = i^2 \cdot i = (-1) \cdot i = -i$.
The condition $\theta \neq -\pi/2$ is given to ensure that Z is well-defined and not purely imaginary negative, but it does not affect the final calculation.