Question Number 212541 by hardmath last updated on 16/Oct/24
$$\mathrm{Find}:\:\:\boldsymbol{\mathrm{x}}\:=\:? \\ $$$$\mathrm{sin}\left(\frac{\mathrm{88}\pi^{\mathrm{2}} }{\mathrm{x}}\right)\:=\:\frac{\mathrm{1}}{\mathrm{cos}\left(\mathrm{3x}\right)} \\ $$
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}](https://www.tinkutara.com/question/Q212546.png)
$$\mathrm{sin}\:\frac{\mathrm{88}\pi^{\mathrm{2}} }{{x}}\:\in\left[−\mathrm{1},\:\mathrm{1}\right] \\ $$$$\frac{\mathrm{1}}{\mathrm{cos}\:\mathrm{3}{x}}\in\left(−\infty,\:−\mathrm{1}\right]\cup\left[\mathrm{1},\:+\infty\right) \\ $$$$\Rightarrow\:\mathrm{cos}\:\mathrm{3}{x}\:=\pm\mathrm{1}\:\Rightarrow\:{x}=\frac{{n}\pi}{\mathrm{3}} \\ $$$$\mathrm{sin}\:\frac{\mathrm{264}\pi}{{n}}\:=\pm\mathrm{1}\:\Rightarrow\:{n}=\frac{\mathrm{528}}{\mathrm{4}{k}\pm\mathrm{1}} \\ $$$$\mathrm{528}=\mathrm{2}^{\mathrm{4}} ×\mathrm{3}×\mathrm{11}\:\mathrm{snd}\:\mathrm{4}{k}\pm\mathrm{1}\:\mathrm{is}\:\mathrm{odd} \\ $$$$\Rightarrow\:\mathrm{4}{k}\pm\mathrm{1}\in\pm\left\{\mathrm{1},\:\mathrm{3},\:\mathrm{11},\:\mathrm{44}\right\} \\ $$$$\Rightarrow\:{n}\in\pm\left\{\mathrm{16},\:\mathrm{48},\:\mathrm{176},\:\mathrm{528}\right\} \\ $$$$\mathrm{but}\:\mathrm{the}\:\mathrm{signs}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sin}\:\mathrm{and}\:\mathrm{cos}\:\mathrm{don}'\mathrm{t}\:\mathrm{always} \\ $$$$\mathrm{match} \\ $$$$\Rightarrow\:{x}\in\left\{−\frac{\mathrm{176}\pi}{\mathrm{3}},\:−\mathrm{16}\pi,\:\frac{\mathrm{16}\pi}{\mathrm{3}},\:\mathrm{176}\pi\right\} \\ $$