2022-06-22

Solve the inequality $\mathrm{sin}\left(x\right)\cdot |\mathrm{tan}x|\le \frac{3}{2}$

Sawyer Day

Yes. the equation is equivalent to the couple of systems:
$\left\{\begin{array}{l}\mathrm{tan}x\ge 0\\ \frac{{\mathrm{sin}}^{2}x}{\mathrm{cos}x}\le \frac{3}{2}\end{array}\phantom{\rule{1em}{0ex}}\vee \phantom{\rule{1em}{0ex}}\left\{\begin{array}{l}\mathrm{tan}x<0\\ -\frac{{\mathrm{sin}}^{2}x}{\mathrm{cos}x}\le \frac{3}{2}\end{array}$

Hint:
$\left\{\begin{array}{llll}\mathrm{tan}x& ,1ºQ& \text{or}& 3ºQ\\ -\mathrm{tan}x& ,2ºQ& \text{or}& 4ºQ\end{array}$
Now solve in each case.
1. $1ºQ$ or $3ºQ$
$3ºQ$
1. $2ºQ$ or $4ºQ$
$-\frac{{\mathrm{sin}}^{2}x}{\mathrm{cos}x}\le \frac{3}{2}\to \frac{-1+{\mathrm{cos}}^{2}x}{\mathrm{cos}x}\le \frac{3}{2}\to \frac{-2+2{\mathrm{cos}}^{2}x-3\mathrm{cos}x}{2\mathrm{cos}x}\le 0$

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