Question Number 147008 by gsk2684 last updated on 17/Jul/21
$${find}\:{the}\:{number}\:{of}\:{values}\:{of}\:\mathrm{cot}\:\theta\: \\ $$$${where}\:\theta\in\left[\frac{\pi}{\mathrm{12}}\:\frac{\pi}{\mathrm{2}}\right]\:{satisfying}\:{the}\: \\ $$$${equation}\:\left[\mathrm{tan}\:\theta.\left[\mathrm{cot}\:\theta\right]\right]=\mathrm{1}\:?\: \\ $$$$\left({where}\:\left[{x}\right]\:{is}\:{greatest}\:{integer}\right. \\ $$$$\left.{less}\:{than}\:{or}\:{equal}\:{to}\:{x}\right) \\ $$
Commented by gsk2684 last updated on 22/Jul/21
$$\left[\mathrm{tan}\:\theta\left[\mathrm{cot}\:\theta\right]\right]=\mathrm{1}\Rightarrow\mathrm{1}\leqslant\mathrm{tan}\:\theta\left[\mathrm{cot}\:\theta\right]<\mathrm{2} \\ $$$$\left[\mathrm{cot}\:\theta\right]\in\left\{\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3}\right\}\:{if}\:\:\theta\in\left[\frac{\pi}{\mathrm{12}}\:\:\frac{\pi}{\mathrm{2}}\right] \\ $$$$\left.{i}\right)\left[\mathrm{cot}\:\theta\right]=\mathrm{1} \\ $$$$\Rightarrow\mathrm{1}\leqslant\mathrm{cot}\:\theta<\mathrm{2}\Rightarrow\mathrm{cot}^{−\mathrm{1}} \mathrm{2}<\theta\leqslant\frac{\pi}{\mathrm{4}} \\ $$$${then}\:\frac{\mathrm{1}}{\mathrm{2}}<\mathrm{tan}\:\theta\leqslant\mathrm{1}\:\&\mathrm{1}\leqslant\mathrm{tan}\:\theta<\mathrm{2} \\ $$$$\Rightarrow\mathrm{tan}\:\theta=\mathrm{1} \\ $$$$\Rightarrow\theta=\frac{\pi}{\mathrm{4}} \\ $$$$\left.{ii}\right)\left[\mathrm{cot}\:\theta\right]=\mathrm{2} \\ $$$$\Rightarrow\mathrm{2}\leqslant\mathrm{cot}\:\theta<\mathrm{3}\Rightarrow\mathrm{cot}^{−\mathrm{1}} \mathrm{3}<\theta\leqslant\mathrm{cot}^{−\mathrm{1}} \mathrm{2} \\ $$$${then}\:\frac{\mathrm{1}}{\mathrm{3}}<\mathrm{tan}\:\theta\leqslant\frac{\mathrm{1}}{\mathrm{2}}\:\&\mathrm{1}\leqslant\mathrm{2}\:\mathrm{tan}\:\theta<\mathrm{2} \\ $$$$\:\frac{\mathrm{1}}{\mathrm{3}}<\mathrm{tan}\:\theta\leqslant\frac{\mathrm{1}}{\mathrm{2}}\:\&\frac{\mathrm{1}}{\mathrm{2}}\leqslant\:\mathrm{tan}\:\theta<\mathrm{1} \\ $$$$\Rightarrow\mathrm{tan}\:\theta=\frac{\mathrm{1}}{\mathrm{2}}\Rightarrow\frac{\pi}{\mathrm{12}}<\theta<\frac{\pi}{\mathrm{4}} \\ $$$$\left.{iii}\right)\left[\mathrm{cot}\:\theta\right]=\mathrm{3} \\ $$$$\Rightarrow\mathrm{3}\leqslant\mathrm{cot}\:\theta<\mathrm{cot}\:\frac{\pi}{\mathrm{12}}\Rightarrow\frac{\pi}{\mathrm{12}}<\theta\leqslant\mathrm{cot}^{−\mathrm{1}} \mathrm{3} \\ $$$${then}\:\mathrm{2}−\sqrt{\mathrm{3}}<\mathrm{tan}\:\theta\leqslant\frac{\mathrm{1}}{\mathrm{3}}\:\&\mathrm{1}\leqslant\mathrm{3}\:\mathrm{tan}\:\theta<\mathrm{2} \\ $$$$\:\mathrm{2}−\sqrt{\mathrm{3}}<\mathrm{tan}\:\theta\leqslant\frac{\mathrm{1}}{\mathrm{3}}\:\&\frac{\mathrm{1}}{\mathrm{3}}\leqslant\:\mathrm{tan}\:\theta<\frac{\mathrm{2}}{\mathrm{3}} \\ $$$$\Rightarrow\mathrm{tan}\:\theta=\frac{\mathrm{1}}{\mathrm{3}}\Rightarrow\frac{\pi}{\mathrm{12}}<\theta<\frac{\pi}{\mathrm{4}} \\ $$$${ans}\:\mathrm{3}\:{values}\:{exist} \\ $$$$ \\ $$