Related-to-Q16675-Find-the-number-of-intersection-points-of-graph-sin-x-x-10-Let-s-see-sin-x-x-n-with-n-gt-1-For-n-1-there-is-one-intersection-point-Let-x-2kpi-t-with-k-N-t-0-2pi-sin-

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Question Number 16699 by mrW1 last updated on 25/Jun/17

$$\mathrm{Related}\mathrm{to}\mathrm{Q}16675:$$$$$$$$\mathrm{Find}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{intersection}\mathrm{points}$$$$\mathrm{of}\mathrm{graph}\sin \mathrm{x}=\frac{\mathrm{x}}{10}.$$$$$$$${\mathrm{Let}}^{\prime }\mathrm{s}\mathrm{see}\sin \mathrm{x}=\frac{\mathrm{x}}{\mathrm{n}}\mathrm{with}\mathrm{n}>1.$$$$\mathrm{For}\mathrm{n}⩽1\mathrm{there}\mathrm{is}\mathrm{one}\mathrm{intersection}\mathrm{point}.$$$$$$$$\mathrm{Let}\mathrm{x}=2\mathrm{k}\pi +\mathrm{t}\mathrm{with}\mathrm{k}\in \mathbb{N}\wedge \mathrm{t}\in [0,2\pi ]$$$$\sin \mathrm{x}=\sin \mathrm{t}$$$$\cos \mathrm{x}=\cos \mathrm{t}$$$$$$$$\mathrm{we}\mathrm{find}\mathrm{the}\mathrm{point}\mathrm{on}\mathrm{f}\left(\mathrm{x}\right)=\sin \mathrm{x}\mathrm{where}\mathrm{its}$$$$\mathrm{tangent}\mathrm{is}\mathrm{g}\left(\mathrm{x}\right)=\frac{\mathrm{x}}{\mathrm{n}}.$$$${\mathrm{f}}^{\prime }\left(\mathrm{x}\right)=\cos \mathrm{x}=\cos \mathrm{t}$$$${\mathrm{g}}^{\prime }\left(\mathrm{x}\right)=\frac{1}{\mathrm{n}}$$$$\cos \mathrm{t}=\frac{1}{\mathrm{n}}$$$$\mathrm{t}={\cos }^{-1}\frac{1}{\mathrm{n}}$$$$\sin \mathrm{t}=\frac{\mathrm{n}}{\sqrt{{\mathrm{n}}^2+1}}$$$$$$$$\mathrm{so}\mathrm{that}\mathrm{f}\left(\mathrm{x}\right)\mathrm{intersects}\mathrm{with}\mathrm{g}\left(\mathrm{x}\right),$$$$\frac{\sin \mathrm{x}}{\mathrm{x}}⩾\frac{1}{\mathrm{n}}$$$$\Rightarrow \mathrm{n}\sin \mathrm{x}⩾\mathrm{x}$$$$\Rightarrow \mathrm{n}\sin \mathrm{t}⩾2\mathrm{k}\pi +\mathrm{t}$$$$\Rightarrow \mathrm{k}⩽\frac{\mathrm{n}\sin \mathrm{t}-\mathrm{t}}{2\pi }=\frac{\frac{{\mathrm{n}}^2}{\sqrt{{\mathrm{n}}^2+1}}-{\cos }^{-1}\frac{1}{\mathrm{n}}}{2\pi }$$$${\mathrm{k}}_{\max }=\lfloor \frac{\frac{{\mathrm{n}}^2}{\sqrt{{\mathrm{n}}^2+1}}-{\cos }^{-1}\frac{1}{\mathrm{n}}}{2\pi }\rfloor$$$$$$$$\mathrm{number}\mathrm{of}\mathrm{intersecting}\mathrm{points}\mathrm{is}$$$$\mathrm{m}=2\times 2({\mathrm{k}}_{\max }+1)-1={4\mathrm{k}}_{\max }+3$$$$$$$$\mathrm{for}\mathrm{n}=10$$$${\mathrm{k}}_{\max }=\lfloor \frac{\frac{{\mathrm{n}}^2}{\sqrt{{\mathrm{n}}^2+1}}-{\cos }^{-1}\frac{1}{\mathrm{n}}}{2\pi }\rfloor$$$$=\lfloor \frac{\frac{{10}^2}{\sqrt{{10}^2+1}}-{\cos }^{-1}\frac{1}{10}}{2\pi }\rfloor =\lfloor 1.35\rfloor =1$$$$\Rightarrow \mathrm{m}=4\times 1+3=7$$$$$$$$\mathrm{for}\mathrm{n}=20$$$${\mathrm{k}}_{\max }=\lfloor \frac{\frac{{20}^2}{\sqrt{{20}^2+1}}-{\cos }^{-1}\frac{1}{20}}{2\pi }\rfloor =\lfloor 2.94\rfloor =2$$$$\Rightarrow \mathrm{m}=4\times 2+3=11$$

Commented by mrW1 last updated on 25/Jun/17

Commented by ajfour last updated on 25/Jun/17

$$\mathrm{good}\mathrm{Sir},\mathrm{and}\mathrm{thank}\mathrm{you}.$$

Commented by Tinkutara last updated on 25/Jun/17

$$\mathrm{Thanks}\mathrm{Sir}!\mathrm{Can}\mathrm{you}\mathrm{give}\mathrm{me}\mathrm{your}$$$$\mathrm{Geometry}\mathrm{Expressions}\mathrm{for}\mathrm{PC}?$$

Commented by mrW1 last updated on 25/Jun/17

$$\mathrm{As}\mathrm{I}\mathrm{have}\mathrm{told}\mathrm{you},\mathrm{you}\mathrm{can}\mathrm{download}$$$$\mathrm{the}\mathrm{software}\mathrm{in}\mathrm{internet}\mathrm{by}\mathrm{yourself}.\mathrm{E}.\mathrm{g}.$$$$\mathrm{https}://\mathrm{dfiles}.\mathrm{eu}/\mathrm{files}/\mathrm{kd}976\mathrm{lf}6\mathrm{v}$$$$\mathrm{It}\mathrm{is}\mathrm{Russian},\mathrm{but}\mathrm{you}\mathrm{can}\mathrm{switch}\mathrm{into}$$$$\mathrm{English}.\mathrm{I}\mathrm{downloaded}\mathrm{it}\mathrm{also}\mathrm{here}.$$

Commented by Tinkutara last updated on 25/Jun/17

$$\mathrm{Sir},\mathrm{this}\mathrm{is}\mathrm{a}.\mathrm{rar}\mathrm{file}\mathrm{and}\mathrm{after}$$$$\mathrm{decompressing}\mathrm{it}\mathrm{I}\mathrm{cannot}\mathrm{find}\mathrm{the}.\mathrm{exe}$$$$\mathrm{file}\mathrm{to}\mathrm{be}\mathrm{run}\mathrm{on}\mathrm{PC}.\mathrm{What}\mathrm{to}\mathrm{do}\mathrm{after}$$$$\mathrm{downloading}?\mathrm{Will}\mathrm{the}\mathrm{application}\mathrm{not}$$$$\mathrm{ask}\mathrm{for}\mathrm{any}\mathrm{type}\mathrm{of}\mathrm{license}\mathrm{or}\mathrm{key}?$$

Commented by mrW1 last updated on 25/Jun/17

$$\mathrm{In}\mathrm{the}\mathrm{rar}\mathrm{file}\mathrm{there}\mathrm{are}2\mathrm{files}:\mathrm{a}.\mathrm{exe}$$$$\mathrm{and}\mathrm{a}.\mathrm{url}\mathrm{file}.\mathrm{You}\mathrm{can}\mathrm{run}\mathrm{the}\mathrm{file}$$$$\mathrm{geometryexpressions}.\mathrm{exe}\mathrm{directly}.$$

Commented by mrW1 last updated on 25/Jun/17

$$\mathrm{You}\mathrm{asked}\mathrm{for}\mathrm{a}\mathrm{version}\mathrm{which}\mathrm{runs}$$$$\mathrm{without}\mathrm{key}.\mathrm{This}\mathrm{is}\mathrm{such}\mathrm{a}\mathrm{version}.$$

Commented by Tinkutara last updated on 25/Jun/17

$$\mathrm{Thanks}\mathrm{a}\mathrm{lot}\mathrm{Sir}!$$

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