Question Number 147008 by gsk2684 last updated on 17/Jul/21
![find the number of values of cot θ where θ∈[(π/(12)) (π/2)] satisfying the equation [tan θ.[cot θ]]=1 ? (where [x] is greatest integer less than or equal to x)](https://www.tinkutara.com/question/Q147008.png)
$${find}\:{the}\:{number}\:{of}\:{values}\:{of}\:\mathrm{cot}\:\theta\: \\ $$$${where}\:\theta\in\left[\frac{\pi}{\mathrm{12}}\:\frac{\pi}{\mathrm{2}}\right]\:{satisfying}\:{the}\: \\ $$$${equation}\:\left[\mathrm{tan}\:\theta.\left[\mathrm{cot}\:\theta\right]\right]=\mathrm{1}\:?\: \\ $$$$\left({where}\:\left[{x}\right]\:{is}\:{greatest}\:{integer}\right. \\ $$$$\left.{less}\:{than}\:{or}\:{equal}\:{to}\:{x}\right) \\ $$
Commented by gsk2684 last updated on 22/Jul/21
![[tan θ[cot θ]]=1⇒1≤tan θ[cot θ]<2 [cot θ]∈{0,1,2,3} if θ∈[(π/(12)) (π/2)] i)[cot θ]=1 ⇒1≤cot θ<2⇒cot^(−1) 2<θ≤(π/4) then (1/2)<tan θ≤1 &1≤tan θ<2 ⇒tan θ=1 ⇒θ=(π/4) ii)[cot θ]=2 ⇒2≤cot θ<3⇒cot^(−1) 3<θ≤cot^(−1) 2 then (1/3)<tan θ≤(1/2) &1≤2 tan θ<2 (1/3)<tan θ≤(1/2) &(1/2)≤ tan θ<1 ⇒tan θ=(1/2)⇒(π/(12))<θ<(π/4) iii)[cot θ]=3 ⇒3≤cot θ<cot (π/(12))⇒(π/(12))<θ≤cot^(−1) 3 then 2−(√3)<tan θ≤(1/3) &1≤3 tan θ<2 2−(√3)<tan θ≤(1/3) &(1/3)≤ tan θ<(2/3) ⇒tan θ=(1/3)⇒(π/(12))<θ<(π/4) ans 3 values exist](https://www.tinkutara.com/question/Q147039.png)
$$\left[\mathrm{tan}\:\theta\left[\mathrm{cot}\:\theta\right]\right]=\mathrm{1}\Rightarrow\mathrm{1}\leqslant\mathrm{tan}\:\theta\left[\mathrm{cot}\:\theta\right]<\mathrm{2} \\ $$$$\left[\mathrm{cot}\:\theta\right]\in\left\{\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3}\right\}\:{if}\:\:\theta\in\left[\frac{\pi}{\mathrm{12}}\:\:\frac{\pi}{\mathrm{2}}\right] \\ $$$$\left.{i}\right)\left[\mathrm{cot}\:\theta\right]=\mathrm{1} \\ $$$$\Rightarrow\mathrm{1}\leqslant\mathrm{cot}\:\theta<\mathrm{2}\Rightarrow\mathrm{cot}^{−\mathrm{1}} \mathrm{2}<\theta\leqslant\frac{\pi}{\mathrm{4}} \\ $$$${then}\:\frac{\mathrm{1}}{\mathrm{2}}<\mathrm{tan}\:\theta\leqslant\mathrm{1}\:\&\mathrm{1}\leqslant\mathrm{tan}\:\theta<\mathrm{2} \\ $$$$\Rightarrow\mathrm{tan}\:\theta=\mathrm{1} \\ $$$$\Rightarrow\theta=\frac{\pi}{\mathrm{4}} \\ $$$$\left.{ii}\right)\left[\mathrm{cot}\:\theta\right]=\mathrm{2} \\ $$$$\Rightarrow\mathrm{2}\leqslant\mathrm{cot}\:\theta<\mathrm{3}\Rightarrow\mathrm{cot}^{−\mathrm{1}} \mathrm{3}<\theta\leqslant\mathrm{cot}^{−\mathrm{1}} \mathrm{2} \\ $$$${then}\:\frac{\mathrm{1}}{\mathrm{3}}<\mathrm{tan}\:\theta\leqslant\frac{\mathrm{1}}{\mathrm{2}}\:\&\mathrm{1}\leqslant\mathrm{2}\:\mathrm{tan}\:\theta<\mathrm{2} \\ $$$$\:\frac{\mathrm{1}}{\mathrm{3}}<\mathrm{tan}\:\theta\leqslant\frac{\mathrm{1}}{\mathrm{2}}\:\&\frac{\mathrm{1}}{\mathrm{2}}\leqslant\:\mathrm{tan}\:\theta<\mathrm{1} \\ $$$$\Rightarrow\mathrm{tan}\:\theta=\frac{\mathrm{1}}{\mathrm{2}}\Rightarrow\frac{\pi}{\mathrm{12}}<\theta<\frac{\pi}{\mathrm{4}} \\ $$$$\left.{iii}\right)\left[\mathrm{cot}\:\theta\right]=\mathrm{3} \\ $$$$\Rightarrow\mathrm{3}\leqslant\mathrm{cot}\:\theta<\mathrm{cot}\:\frac{\pi}{\mathrm{12}}\Rightarrow\frac{\pi}{\mathrm{12}}<\theta\leqslant\mathrm{cot}^{−\mathrm{1}} \mathrm{3} \\ $$$${then}\:\mathrm{2}−\sqrt{\mathrm{3}}<\mathrm{tan}\:\theta\leqslant\frac{\mathrm{1}}{\mathrm{3}}\:\&\mathrm{1}\leqslant\mathrm{3}\:\mathrm{tan}\:\theta<\mathrm{2} \\ $$$$\:\mathrm{2}−\sqrt{\mathrm{3}}<\mathrm{tan}\:\theta\leqslant\frac{\mathrm{1}}{\mathrm{3}}\:\&\frac{\mathrm{1}}{\mathrm{3}}\leqslant\:\mathrm{tan}\:\theta<\frac{\mathrm{2}}{\mathrm{3}} \\ $$$$\Rightarrow\mathrm{tan}\:\theta=\frac{\mathrm{1}}{\mathrm{3}}\Rightarrow\frac{\pi}{\mathrm{12}}<\theta<\frac{\pi}{\mathrm{4}} \\ $$$${ans}\:\mathrm{3}\:{values}\:{exist} \\ $$$$ \\ $$