Question Number 48851 by vajpaithegrate@gmail.com last updated on 29/Nov/18 | ||
$$\mathrm{the}\:\mathrm{total}\:\mathrm{no}\:\mathrm{of}\:\mathrm{solution}\:\mathrm{of}\:\mathrm{sin}\left\{\mathrm{x}\right\}=\mathrm{cos}\left\{\mathrm{x}\right\} \\ $$$$\mathrm{where}\left\{\mathrm{x}\right\}\:\mathrm{denotes}\:\mathrm{fractional}\:\mathrm{part}\:\mathrm{of}\:\mathrm{x}\:\mathrm{in} \\ $$$$\left[\mathrm{o}\:\mathrm{2}\pi\right]\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\mathrm{ans}:\mathrm{6} \\ $$ | ||
Commented by tanmay.chaudhury50@gmail.com last updated on 29/Nov/18 | ||
$${sin}\left\{{x}\right\}={cos}\left\{{x}\right\} \\ $$$${sin}\left({x}−\left[{x}\right]\right)={cos}\left({x}−\left[{x}\right]\right) \\ $$$${x}=\left\{{x}\right\}+\left[{x}\right] \\ $$$$\left\{{x}\right\}={fractional}\:{part}\:{function} \\ $$$$\left[{x}\right]={greatest}\:{integer}\:{function} \\ $$$${from}\:{graph}\:\:{it}\:{is}\:{seen}\:{that}\:{in}\:\left[\mathrm{0},\mathrm{2}\pi\right] \\ $$$${six}\left(\mathrm{06}\right)\:{solution} \\ $$ | ||
Commented by tanmay.chaudhury50@gmail.com last updated on 29/Nov/18 | ||
Commented by tanmay.chaudhury50@gmail.com last updated on 29/Nov/18 | ||
$${graph}\:{of}\:{sin}\left\{{x}\right\} \\ $$$${sin}\left({x}−\left[{x}\right]\right. \\ $$ | ||
Commented by tanmay.chaudhury50@gmail.com last updated on 29/Nov/18 | ||
Commented by tanmay.chaudhury50@gmail.com last updated on 29/Nov/18 | ||
$${graph}\:{of}\:{cos}\left\{{x}\right\} \\ $$$${graph}\:{of}\:{cos}\left({x}−\left[{x}\right]\right) \\ $$ | ||
Commented by tanmay.chaudhury50@gmail.com last updated on 29/Nov/18 | ||
Commented by tanmay.chaudhury50@gmail.com last updated on 29/Nov/18 | ||
$$\mathrm{2}\pi=\mathrm{2}×\mathrm{3}.\mathrm{14}=\mathrm{6}.\mathrm{28} \\ $$$${in}\:{graph}\:\:{it}\:{is}\:{seen}\:{that}\:{from}\:{x}=\mathrm{0}\:{to}\:{x}=\mathrm{6} \\ $$$${there}\:{are}\:{six}\:{point}\:{where}\:{sin}\left\{{x}\right\}={cos}\left\{{x}\right\} \\ $$$${intersects}.\:\:{so}\:{six}\:{solution}\:{in}\:{x}\:\left(\mathrm{0},\mathrm{2}\pi\right) \\ $$ | ||
Commented by vajpaithegrate@gmail.com last updated on 29/Nov/18 | ||
$$\mathrm{tnq}\:\mathrm{sir} \\ $$ | ||
Commented by vajpaithegrate@gmail.com last updated on 29/Nov/18 | ||
$$\mathrm{any}\:\mathrm{alternate}\:\mathrm{method}, \\ $$ | ||
Answered by mr W last updated on 29/Nov/18 | ||
$${let}\:{t}=\left\{{x}\right\} \\ $$$$\mathrm{0}\leqslant{t}<\mathrm{1} \\ $$$$\mathrm{sin}\:{t}=\mathrm{cos}\:{t} \\ $$$$\Rightarrow\mathrm{tan}\:{t}=\mathrm{1} \\ $$$$\Rightarrow{t}=\pi/\mathrm{4}=\mathrm{0}.\mathrm{785} \\ $$$$\Rightarrow{solution}\:{is}\:{x}={n}+{t}\:{with}\:{n}=\mathrm{0},\mathrm{1},\mathrm{2},.... \\ $$$${i}.{e}.\:{x}=\mathrm{0}.\mathrm{785},\:\mathrm{1}.\mathrm{785},\:\mathrm{2}.\mathrm{785},...,\mathrm{5}.\mathrm{785},\:\mathrm{6}.\mathrm{785},...... \\ $$$$\mathrm{2}\pi=\mathrm{6}.\mathrm{283}<\mathrm{6}.\mathrm{785} \\ $$$$\Rightarrow\mathrm{6}\:{solutions}\:{within}\:\left[\mathrm{0},\:\mathrm{2}\pi\right]: \\ $$$${x}=\mathrm{0}.\mathrm{785},\:\mathrm{1}.\mathrm{785},\:\mathrm{2}.\mathrm{785},...,\mathrm{5}.\mathrm{785} \\ $$ | ||
Commented by vajpaithegrate@gmail.com last updated on 29/Nov/18 | ||
$$\mathrm{tnq}\:\mathrm{sir} \\ $$ | ||
Commented by tanmay.chaudhury50@gmail.com last updated on 29/Nov/18 | ||
$${excellent}\:{sir}... \\ $$ | ||