Find-x-sin-88pi-2-x-1-cos-3x-

Question Number 212541 by hardmath last updated on 16/Oct/24

Find : x = ?

\sin (\frac{88 π^{2}}{x}) = \frac{1}{\cos (3 x)}

Answered by Frix last updated on 17/Oct/24

\sin \frac{88 π^{2}}{x} \in [- 1, 1]

\frac{1}{\cos 3 x} \in (- \infty, - 1] \cup [1, + \infty)

\Rightarrow \cos 3 x = \pm 1 \Rightarrow x = \frac{n π}{3}

\sin \frac{264 π}{n} = \pm 1 \Rightarrow n = \frac{528}{4 k \pm 1}

528 = 2^{4} \times 3 \times 11 snd 4 k \pm 1 is odd

\Rightarrow 4 k \pm 1 \in \pm {1, 3, 11, 44}

\Rightarrow n \in \pm {16, 48, 176, 528}

but the signs of the \sin and \cos {don}^{'} t always

match

\Rightarrow x \in {- \frac{176 π}{3}, - 16 π, \frac{16 π}{3}, 176 π}

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