First slide

Algebra of complex numbers

Question

The complex numbers sin x + i sin 2x and cos x - i sin 2x are conjugate to each other, for

Moderate

A

$x = nπ, n \in Z$
B

x = 0
C

$x = (n + 1 / 2) π, n \in Z$
D

no value of x

Solution

Let $z_{1} = \sin x + icos 2 x and z_{2} = \cos x - isin 2 x$ .
Then ${\bar{z}}_{1} = z_{2}$
$\begin{array}{l} \Rightarrow \sin x - icos 2 x = \cos x - isin 2 x \\ \Rightarrow \sin x = \cos x and \cos 2 x = \sin 2 x \\ \Rightarrow \tan x = 1 and \tan 2 x = 1 \\ \Rightarrow x = \frac{π}{4} and x = \frac{π}{8} \end{array}$
This is not possible.
Hence, there is no value of x.

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