Question

Find the values of $ x $ for which the expressions $$ \sin x + i\cos 2x \quad \text{and} \quad \cos x - i\sin 2x $$ are conjugate to each other.

• A. \( x = n\pi \pm \frac{\pi}{6} \)
• B. None
• C. \( x = n\pi \pm \frac{\pi}{3} \)
• D. \( x = \left(n + \frac{1}{2} \right)\pi \)

Hint:

For two expressions to be conjugates, match real parts and set imaginary parts equal but opposite in sign.

Answer 1

The Correct Option is B

Two complex numbers \( z_1 = a + ib \) and \( z_2 = c + id \) are conjugates iff: \[ z_2 = \overline{z_1} = a - ib \] Given: \[ z_1 = \sin x + i\cos 2x, \quad z_2 = \cos x - i\sin 2x \] For them to be conjugates: \[ \cos x - i\sin 2x = \sin x - i\cos 2x \Rightarrow \cos x = \sin x,\quad \sin 2x = \cos 2x \] From \( \cos x = \sin x \Rightarrow x = \frac{\pi}{4} + n\pi \)
Now check \( \sin 2x = \cos 2x \Rightarrow 2x = \frac{\pi}{4} + m\pi \Rightarrow x = \frac{\pi}{8} + \frac{m\pi}{2} \)
These values do not overlap, hence no solution satisfies both conditions simultaneously.