First slide

Algebra of complex numbers

Question

If $\frac{3 + i \sin θ}{4 - i \cos θ}, θ \in [0, 2 π]$ , is a real number, then an argument of $\sin θ + i \cos θ$ is:

Easy

A

$- \tan^{- 1} (\frac{3}{4})$
B

$π - \tan^{- 1} (\frac{3}{4})$
C

$π - \tan^{- 1} (\frac{4}{3})$
D

$\tan^{- 1} (\frac{4}{3})$

Solution

$z = \frac{(3 + i \sin θ)}{(4 - i \cos θ)} \times \frac{(4 + i \cos θ)}{4 + i \cos θ}$
as z is purely real $\Rightarrow 3 \cos θ + 4 \sin θ = 0 \Rightarrow \tan θ = - \frac{3}{4}$
$∴ \arg (\sin θ + i \cos θ) = π - \tan^{- 1} |\frac{\cos θ}{\sin θ}| o r - \tan^{- 1} |\frac{\cos θ}{\sin θ}| = π - \tan^{- 1} (\frac{4}{3}) o r - \tan^{- 1} (\frac{4}{3})$

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