Find: x = ? sin(((88π^2 )/x)) = (1/(cos(3x)))


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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
	$sin ((88π^2 )/x) ∈[−1, 1] (1/(cos 3x))∈(−∞, −1]∪[1, +∞) ⇒ cos 3x =±1 ⇒ x=((nπ)/3) sin ((264π)/n) =±1 ⇒ n=((528)/(4k±1)) 528=2^4 ×3×11 snd 4k±1 is odd ⇒ 4k±1∈±{1, 3, 11, 44} ⇒ n∈±{16, 48, 176, 528} but the signs of the sin and cos don′t always match ⇒ x∈{−((176π)/3), −16π, ((16π)/3), 176π}$
	$\sin \frac{88 π^{2}}{x} \in [- 1, 1]$ $\frac{1}{\cos 3 x} \in (- \infty, - 1] \cup [1, + \infty)$ $\Rightarrow \cos 3 x = \pm 1 \Rightarrow x = \frac{n π}{3}$ $\sin \frac{264 π}{n} = \pm 1 \Rightarrow n = \frac{528}{4 k \pm 1}$ $528 = 2^{4} \times 3 \times 11 snd 4 k \pm 1 is odd$ $\Rightarrow 4 k \pm 1 \in \pm {1, 3, 11, 44}$ $\Rightarrow n \in \pm {16, 48, 176, 528}$ $but the signs of the \sin and \cos {don}^{'} t always$ $match$ $\Rightarrow x \in {- \frac{176 π}{3}, - 16 π, \frac{16 π}{3}, 176 π}$
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