First slide

Functions (XII)

Question

let $f (θ) = |\begin{array}{ccc} \sin^{2} θ & \cos^{2} θ & 1 + 4 \sin 4 θ \\ \sin^{2} θ & 1 + \cos^{2} θ & 4 \sin 4 θ \\ 1 + \sin^{2} θ & \cos^{2} θ & 4 \sin 4 θ \end{array}|$ then f is

Moderate

A

a non periodic function
B

periodic with period $π$
C

periodic with period $\frac{π}{2}$
D

odd function

Solution

$f (θ) = |\begin{array}{ccc} - 1 & 0 & 1 \\ - 1 & 1 & 0 \\ 1 + \sin^{2} θ & \cos^{2} θ & 4 \sin 2 θ \end{array}|$
$(R_{1} \to R_{1} - R_{3} R_{2} \to R_{2} - R_{3})$
$= |\begin{array}{ccc} 0 & 0 & 1 \\ - 1 & 1 & 0 \\ 1 + \sin^{2} θ & \cos^{2} θ & 4 \sin 4 θ \\ + 4 \sin 4 θ \end{array}| (C 1 \to C 1 + C 3)$
$= - (\cos^{2} θ + 1 + \sin^{2} θ + 4 \sin 4 θ)$
$\Rightarrow - 2 (1 + 2 \sin 4 θ)$
which is periodic function with period $\frac{2 π}{4} = \frac{π}{2}$ .

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