The number of elements in the set $$S = \{ \theta \in [ - 4\pi ,4\pi ]:3{\cos ^2}2\theta + 6\cos 2\theta - 10{\cos ^2}\theta + 5 = 0\} $$ is __________.
Answer (integer)
32
Solution
$3 \cos ^{2} 2 \theta+6 \cos 2 \theta-\frac{10(1+\cos 2 \theta)}{2}+5=0$
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$\Rightarrow 3 \cos ^{2} 2 \theta+\cos 2 \theta=0$
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$\Rightarrow \cos 2 \theta=0$ or $\cos 2 \theta=\frac{-1}{3}$
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As $\theta \in[0, \pi], \cos 2 \theta=\frac{-1}{3} \Rightarrow 2$ times
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$\Rightarrow \theta \in[-4 \pi, 4 \pi], \cos 2 \theta=\frac{-1}{3} \Rightarrow 16$ times
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Similarly, $\cos 2 \theta=0 \Rightarrow 16$ times
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$\therefore $ Total 32 solutions
About this question
Subject: Mathematics · Chapter: Trigonometric Functions · Topic: Trigonometric Ratios and Identities
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