Question

If \(\cos\theta + i \sin\theta, \, \theta \in \mathbb{R}\), is a root of the equation
\[ a_0 x^n + a_1 x^{n-1} + \cdots + a_{n-1}x + a_n = 0, \, a_0, a_1, \ldots, a_n \in \mathbb{R}, \, a_0 \neq 0 \]
then the value of \(a_1 \sin\theta + a_2 \sin 2\theta + \cdots + a_n \sin n\theta\) is:

• A. \(2n\)
• B. \(n\)
• C. \(0\)
• D. \(n + 1\)

Hint:

When dealing with roots of complex numbers in polynomial equations, use Euler’s formula to convert the trigonometric terms into exponential form and simplify the equation.