The sum of all values of [ 0 , 2 ] satisfying 2 sin 2 = cos 2 and 2 cos 2 = 3 sin is: [2025]

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Solution

Q.
The sum of all values of $θ \in [0, 2 π]$ satisfying $2 \sin^{2} θ = \cos 2 θ$ and $2 \cos^{2} θ = 3 \sin θ$ is: [2025]

1 $π / 2$

2 $π$

3 $5 π / 6$

4 $4 π$

Ans.
(2)

$2 \sin^{2} θ = \cos 2 θ$

$\Rightarrow 2 \sin^{2} θ = 1 - 2 \sin^{2} θ \Rightarrow 4 \sin^{2} θ = 1$

$\Rightarrow \sin^{2} θ = \frac{1}{4} \Rightarrow \sin θ = \pm \frac{1}{2}$ ... (i)

Also, $2 \cos^{2} θ = 3 \sin θ$

$\Rightarrow 2 (1 - \sin^{2} θ) = 3 \sin θ$

$\Rightarrow 2 \sin^{2} θ + 3 \sin θ - 2 = 0$

$\Rightarrow 2 \sin^{2} θ + 4 \sin θ - \sin θ - 2 = 0$

$\Rightarrow (2 \sin θ - 1) (\sin θ + 2) = 0$

$\Rightarrow \sin θ = \frac{1}{2} (\sin θ = - 2 (not possible))$ ... (ii)

Thus, $θ = \frac{π}{6}, \frac{5 π}{6} (θ \in [0, 2 π])$ [Using (i) & (ii)]

$∴$ Required sum = $π$

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