First slide

Algebra of complex numbers

Question

$\frac{5 + i \sin θ}{5 - 3 i \sin θ}$ is a real number when

Moderate

A

$θ = π / 4$
B

$θ = - π$
C

$θ = - π / 2$
D

$θ = π / 2$

Solution

$\frac{5 + i \sin θ}{5 - 3 i \sin θ}$ is a real number
$\Leftrightarrow \frac{5 + i \sin θ}{5 - 3 i \sin θ} = \frac{5 - i \sin θ}{5 + 3 i \sin θ}$
$\Leftrightarrow (5 + i \sin θ) (5 + 3 i \sin θ)$
$= (5 - 3 i \sin θ) (5 - i \sin θ)$
$= 25 - 20 i \sin θ - 3 \sin^{2} θ$
$\Leftrightarrow 40 i \sin θ = 0 \Leftrightarrow \sin θ = 0$
This is possible when $θ = - π$

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