Question and Answers Forum

All Questions      Topic List

None Questions

Previous in All Question      Next in All Question      

Previous in None      Next in None      

Question Number 193494 by aba last updated on 15/Jun/23

Let n be a fixed positive integer such that sin((Ο€/(2n)))+cos((𝛑/(2n)))=((√n)/2) Then find n

$$\boldsymbol{\mathrm{Let}}\:\boldsymbol{\mathrm{n}}\:\boldsymbol{\mathrm{be}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{fixed}}\:\boldsymbol{\mathrm{positive}}\:\boldsymbol{\mathrm{integer}}\:\boldsymbol{\mathrm{such}}\:\boldsymbol{\mathrm{that}} \\ $$$$\mathrm{sin}\left(\frac{\pi}{\mathrm{2n}}\right)+\mathrm{cos}\left(\frac{\boldsymbol{\pi}}{\mathrm{2n}}\right)=\frac{\sqrt{\mathrm{n}}}{\mathrm{2}} \\ $$$$\boldsymbol{\mathrm{Then}}\:\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{n}} \\ $$

Commented by maths_plus last updated on 15/Jun/23

cool

$$\mathrm{cool} \\ $$

Answered by Frix last updated on 15/Jun/23

n=6

$${n}=\mathrm{6} \\ $$

Answered by witcher3 last updated on 15/Jun/23

∣cos(a)+sin(a)∣=(√(2⟨∣))sin(a+(Ο€/4))βˆ£β‰€(√2) β‡’((√n)/2)≀(√2)β‡’n≀8 n∈{1,....8} (sin((Ο€/(2n)))+cos((Ο€/(2n))))^2 =1+sin((Ο€/n))=(n/4) sin((Ο€/n))=((nβˆ’4)/4)..E sin(x)≀xβ‡’((nβˆ’4)/4)≀(Ο€/n)≀(4/n)β‡’n^2 βˆ’4nβˆ’16≀0 n∈{((4βˆ’(√(80)))/2),((4+(√(80)))/2)}β‡’n≀((13)/2)β‡’n≀6 sin n∈{1....8}β‡’sin((Ο€/n))>0..Eβ‡’nβˆ’4>0β‡’nβ‰₯5 n∈{5,6} simply check n=6 unique solution

$$\mid\mathrm{cos}\left(\mathrm{a}\right)+\mathrm{sin}\left(\mathrm{a}\right)\mid=\sqrt{\mathrm{2}\langle\mid}\mathrm{sin}\left(\mathrm{a}+\frac{\pi}{\mathrm{4}}\right)\mid\leqslant\sqrt{\mathrm{2}} \\ $$$$\Rightarrow\frac{\sqrt{\mathrm{n}}}{\mathrm{2}}\leqslant\sqrt{\mathrm{2}}\Rightarrow\mathrm{n}\leqslant\mathrm{8} \\ $$$$\mathrm{n}\in\left\{\mathrm{1},....\mathrm{8}\right\} \\ $$$$\left(\mathrm{sin}\left(\frac{\pi}{\mathrm{2n}}\right)+\mathrm{cos}\left(\frac{\pi}{\mathrm{2n}}\right)\right)^{\mathrm{2}} =\mathrm{1}+\mathrm{sin}\left(\frac{\pi}{\mathrm{n}}\right)=\frac{\mathrm{n}}{\mathrm{4}} \\ $$$$\mathrm{sin}\left(\frac{\pi}{\mathrm{n}}\right)=\frac{\mathrm{n}βˆ’\mathrm{4}}{\mathrm{4}}..\mathrm{E} \\ $$$$\mathrm{sin}\left(\mathrm{x}\right)\leqslant\mathrm{x}\Rightarrow\frac{\mathrm{n}βˆ’\mathrm{4}}{\mathrm{4}}\leqslant\frac{\pi}{\mathrm{n}}\leqslant\frac{\mathrm{4}}{\mathrm{n}}\Rightarrow\mathrm{n}^{\mathrm{2}} βˆ’\mathrm{4n}βˆ’\mathrm{16}\leqslant\mathrm{0} \\ $$$$\mathrm{n}\in\left\{\frac{\mathrm{4}βˆ’\sqrt{\mathrm{80}}}{\mathrm{2}},\frac{\mathrm{4}+\sqrt{\mathrm{80}}}{\mathrm{2}}\right\}\Rightarrow\mathrm{n}\leqslant\frac{\mathrm{13}}{\mathrm{2}}\Rightarrow\mathrm{n}\leqslant\mathrm{6} \\ $$$$\mathrm{sin}\:\mathrm{n}\in\left\{\mathrm{1}....\mathrm{8}\right\}\Rightarrow\mathrm{sin}\left(\frac{\pi}{\mathrm{n}}\right)>\mathrm{0}..\mathrm{E}\Rightarrow\mathrm{n}βˆ’\mathrm{4}>\mathrm{0}\Rightarrow\mathrm{n}\geqslant\mathrm{5} \\ $$$$\mathrm{n}\in\left\{\mathrm{5},\mathrm{6}\right\} \\ $$$$\mathrm{simply}\:\mathrm{check}\:\mathrm{n}=\mathrm{6}\:\mathrm{unique}\:\mathrm{solution} \\ $$$$ \\ $$

Commented by York12 last updated on 16/Jun/23

Witcher3 are you using discord

$${Witcher}\mathrm{3}\:{are}\:{you}\:{using}\:{discord} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Answered by MM42 last updated on 15/Jun/23

1+sin((Ο€/n))=(n/4)β‡’sin((Ο€/n))=((nβˆ’4)/4) 0≀(n/4)βˆ’1≀1β‡’4≀n≀8 only n=6 there is a tie

$$\mathrm{1}+{sin}\left(\frac{\pi}{{n}}\right)=\frac{{n}}{\mathrm{4}}\Rightarrow{sin}\left(\frac{\pi}{{n}}\right)=\frac{{n}βˆ’\mathrm{4}}{\mathrm{4}} \\ $$$$\mathrm{0}\leqslant\frac{{n}}{\mathrm{4}}βˆ’\mathrm{1}\leqslant\mathrm{1}\Rightarrow\mathrm{4}\leqslant{n}\leqslant\mathrm{8} \\ $$$${only}\:\:\:{n}=\mathrm{6}\:\:{there}\:{is}\:{a}\:{tie} \\ $$$$ \\ $$

Commented by aba last updated on 15/Jun/23

����

Terms of Service

Privacy Policy

Contact: info@tinkutara.com