The number of elements in the set S = [ 0 , 2 ] : 3 cos 4 - 5 cos 2 - 2 sin 6 + 2 = 0 is [2023]

Question

The number of elements in the set $S = {θ \in [0, 2 π] : 3 \cos^{4} θ - 5 \cos^{2} θ - 2 \sin^{6} θ + 2 = 0}$ is [2023]

Smart Achievers · Accepted Answer

(3)

$3 \cos^{4} θ - 5 \cos^{2} θ - 2 \sin^{6} θ + 2 = 0$

$\Rightarrow 3 \cos^{4} θ - 3 \cos^{2} θ - 2 \cos^{2} θ - 2 \sin^{6} θ + 2 = 0$

$\Rightarrow 3 \cos^{4} θ - 3 \cos^{2} θ + 2 \sin^{2} θ - 2 \sin^{6} θ = 0$

$\Rightarrow 3 \cos^{2} θ (\cos^{2} θ - 1) - 2 \sin^{2} θ (\sin^{4} θ - 1) = 0$

$\Rightarrow - 3 \cos^{2} θ \sin^{2} θ + 2 \sin^{2} θ (1 + \sin^{2} θ) \cos^{2} θ = 0$

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

$\Rightarrow \sin^{2} θ \cos^{2} θ (2 \sin^{2} θ - 1) = 0$

Case I: $\sin^{2} θ = 0 \to 3 solutions: θ = {0, π, 2 π}$

Case II: $\cos^{2} θ = 0 \to 2 solutions: θ = {\frac{π}{2}, \frac{3 π}{2}}$

Case III: $\sin^{2} θ = \frac{1}{2} \to 4 solutions: θ = {\frac{π}{4}, \frac{3 π}{4}, \frac{5 π}{4}, \frac{7 π}{4}}$

$Number of solutions = 9$

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