First slide

Trigonometric equations

Question

A value of $θ$ lying between $θ = 0$ and $θ = π / 2$ and satisfying the equation $|\begin{array}{ccc} 1 + \sin^{2} θ & \cos^{2} θ & 4 \sin 4 θ \\ \sin^{2} θ & 1 + \cos^{2} θ & 4 \sin 4 θ \\ \sin^{2} θ & \cos^{2} θ & 1 + 4 \sin 4 θ \end{array}| = 0$ is

Moderate

A

$3 π / 24$
B

$5 π / 24$
C

$11 π / 24$
D

$π / 24$

Solution

Applying $R_{1} \to R_{1} - R_{3}, R_{2} \to R_{2} - R_{3}$ to the given determinant we get
$|\begin{array}{ccc} 1 & 0 & - 1 \\ 0 & 1 & - 1 \\ \sin^{2} θ & \cos^{2} θ & 1 + 4 \sin 4 θ \end{array}| = 0$
$\begin{array}{l} \Rightarrow 1 + 4 \sin 4 θ + \cos^{2} θ + \sin^{2} θ = 0 \end{array}$
$\Rightarrow \sin 4 θ = - 1 / 2 \Rightarrow 4 θ = π + π / 6$ or $2 π - π / 6$ $[∵ 0 < 4 θ < 2 π]$
$\Rightarrow θ = 7 π / 24$ or $11 π / 24$

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