First slide

Algebra of complex numbers

Question

The number of values of $θ \in (0, π]$ , such that $(\cos θ + i \sin θ) (\cos 3 θ + i \sin 3 θ) (\cos 5 θ + i \sin 5 θ) (\cos 7 θ + i \sin 7 θ) (\cos 9 θ + i \sin 9 θ) = - 1$
is

Moderate

A

11
B

13
C

14
D

16

Solution

Using $\begin{aligned} (\cos α + i \sin α) (\cos β + i \sin β) \\ = \cos (α + β) + i \sin (α + β) \end{aligned}$
We get
$\begin{aligned} \cos (25 θ) = - 1, \sin (25 θ) = 0 \\ \Rightarrow 25 θ = (2 k + 1) π, k \in I . \end{aligned}$
Now, $\begin{aligned} 0 < \frac{(2 k + 1) π}{25} \leq π \\ \Rightarrow 1 \leq (2 k + 1) \leq 25 \\ \Rightarrow k = 0, 1, 2, \dots, 12 \end{aligned}$

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