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If \[\displaystyle{{ f }_{ n }(x)={ \cos ^{ n }{ x } +\cos ^{ n }{ (x+\cfrac { 2\pi }{ 3 } ) } +\cos ^{ n }{ (x+\cfrac { 4\pi }{ 3 } ) } }}\]. Then Solve for x if \(\displaystyle{{ f }_{ 7 }(x)=0}\).


Use Any Tool of Mathematics. Can You Guess why I choose \(\displaystyle{\cfrac { 2\pi }{ 3 } \& \cfrac { 4\pi }{ 3 } }\). ?

Note by Deepanshu Gupta
2 years, 4 months ago

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\( y = cosx + isinx\)

\(\displaystyle{y^{ n }+\cfrac { 1 }{ y^{ n } } =2cosnx=(cosx+isinx)^{ n }=cosnx+isinnx}\).

\(y + \dfrac{1}{y} = 2cosx\)

\( cos^{7}x = ( y + \dfrac{1}{y})^{7}\)

\( = y^{7} + 7y^{5} + 21y^{3} + 35y + 35\dfrac{1}{y} + 21\dfrac{1}{y^{3}} + 7\dfrac{1}{y^{5}} + \dfrac{1}{y^{7}}\)

\( = ( y^{7} + \dfrac{1}{y^{7}}) + 7(y^{5} + \dfrac{1}{y^{5}}) + 21(y^{3} + \dfrac{1}{y^{3}}) + 35(y + \dfrac{1}{y})\)

\( = 2cos7x + 7.2cos5x + 21.2cos3x + 35.2cosx\)

\( cos^{7}x = \dfrac{1}{64}(cos7x + 7cos5x + 21cos3x + 35cosx)\)


\(cos^7(x+ \dfrac{2\pi}{3}) = \dfrac{1}{64}( - cos(7x - \dfrac{\pi}{3}) - 7cos(5x + \dfrac{\pi}{3}) + 21cos3x - 35cos(x - \dfrac{\pi}{3}))\)

\( cos^7(x + \dfrac{4\pi}{3}) = \dfrac{1}{64}( - cos(7x + \dfrac{\pi}{3}) - 7cos(5x - \dfrac{\pi}{3}) + 21cos3x - 35cos(x + \dfrac{\pi}{3}))\)


\( cosA + cosB = 2cos(\dfrac{A + B}{2})cos(\dfrac{A - B}{2})\)

\(cos^7(x+ \dfrac{2\pi}{3}) + ( cos^7(x + \dfrac{4\pi}{3}) = \dfrac{1}{64}( - cos7x - 7cos5x + 42cos3x - 35cosx)\)

\( Expression = \dfrac{63}{64}cos3x = 0 =cos\dfrac{\pi}{2}\)

\( 3x = n\pi \pm \dfrac{\pi}{2}\)

\( x = \dfrac{n\pi}{3} - \dfrac{\pi}{6}\) Megh Choksi · 2 years, 4 months ago

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@Megh Choksi Nice Megh , I'm Expecting It from You :) . But Still There is one more way of similar Type ! Just Focus on the the angles given in question .I beleive that You will Definitely Get it , just give one more Try !\(\ddot\smile\) Deepanshu Gupta · 2 years, 4 months ago

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do we have to use binomial expanision In the mid of the solution . plz ans my question............ and reply Rajat Kharbanda · 2 years, 4 months ago

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@Rajat Kharbanda Yes We Have To use Binomial , But Don't Need to write whole Terms , By Sharp Observation we see that it will cancel out , You Have To Check only Those Terms which are independent of w and w^2 (By using General term for binomial) Deepanshu Gupta · 2 years, 4 months ago

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why Complex Numbers Tag ? :O Karan Shekhawat · 2 years, 4 months ago

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@Karan Shekhawat See megh solution , you will find it interesting :) Deepanshu Gupta · 2 years, 4 months ago

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\((e^{ix} + e^{-ix})/2 = cosx\) given equatin = 0 this implies, \(((e^{ix} + e^{-ix})/2)^{7} (1+w^2 +w) =0\) this implies for all x...... Rajat Kharbanda · 2 years, 4 months ago

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@Rajat Kharbanda Not getting for other terms, please elaborate Megh Choksi · 2 years, 4 months ago

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@Rajat Kharbanda Great ! You Catch Right Path , But You r doing calculation mistake , Infact You can't able To take common \((1+w^2 +w)\) Check ur calculation carefully that wheather you are writing \(w\) or \(w^2\). Deepanshu Gupta · 2 years, 4 months ago

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@Deepanshu Gupta its a long question do we have to apply binomial theorem(to expand) . Actuslly I got the answer which megh did. GREAT QUESTION. WHERE DID YOU GET IT FROM. and where u wake up night at 2:00 clock........................................ Rajat Kharbanda · 2 years, 4 months ago

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@Rajat Kharbanda You Don't Need To calculate whole Terms , They will Cancel out and Only Thing left is , \(_{ 2 }^{ 7 }{ C }{ e }^{ i(3x) }\quad +\quad _{ 5 }^{ 7 }{ C }{ e }^{ -i(3x) }=0\).

And what do you mean by "where u wke up night at 2.00 clock" ? Sorry But I didn't understand it. Deepanshu Gupta · 2 years, 4 months ago

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@Deepanshu Gupta no, you posted in 200 clock night , were you woken up. just didn't mind.. could you try the questin I have posted above .... URGENT... Rajat Kharbanda · 2 years, 4 months ago

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@Rajat Kharbanda Actually I ddin't Sleep , Beacuese I Used to take Sleep in Day- Evening , So I Sleep in Late Night :) Deepanshu Gupta · 2 years, 4 months ago

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Note that due to the the \(2\pi\) periodicity of cosine the given problem is the same as:

\( \cos^7{x}+\cos^7{(x+\frac{2\pi}{3})}+\cos^7{(x-\frac{2\pi}{3})}=0 \)

If only \(\cos^7{x}\) were an odd function around \(x\) the last two terms would cancel out! Note that at \(\frac{\pi}{2}\), this is the case: cosine becomes odd when shifted by \(\frac{\pi}{2}\).. But it just so happens that substituting \(\frac{\pi}{2}\) kills the \(\cos^7{x}\) term as well! Therefore, \(x = \frac{\pi}{2}\) is a solution. Roy Tu · 2 years, 4 months ago

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@Roy Tu wrong ! You Calculate only one Solution ! i.e pi/2 Satisfy , But it in not only solution Recheck ur Thinking (Terms will not cancel out) Deepanshu Gupta · 2 years, 4 months ago

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@Deepanshu Gupta Same thing happens at every increment of \(\pi\) after that, so \(\frac{\pi}{2}+n\pi\) for all integers \(n\). I don't think that's an exhaustive set of solutions, might come back to this tomorrow. Roy Tu · 2 years, 4 months ago

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solve this: Click Here Visakh Radhakrishnan · 2 years, 4 months ago

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@Visakh Radhakrishnan It is not Clearly Stated ,

But As far as I can compile This question I'am getting , velocity as a function of theta as: \[\displaystyle{{ v }=\sqrt { \cfrac { 2Rg }{ 4{ \mu }^{ 2 }+1 } ((2{ { \mu } }^{ 2 }+1)\cos { \theta } -{ \mu }\sin { \theta } -(2{ { \mu } }^{ 2 }+1){ e }^{ -2\mu \theta }) } }\].

and \[\displaystyle{{ v }_{ max }=\sqrt { Rg(\mu \cos { \theta } -\sin { \theta } ) } }\].

Now Putting This in velocity function we should get \(\theta\) and Then We calculate distance By using fact that

\[S=R\theta \].

But it is Too nasty ! Am I correctly understand ur Question ?

And Seriously It is Level-3 ??

@Visakh Radhakrishnan Deepanshu Gupta · 2 years, 4 months ago

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@Deepanshu Gupta maybe he means to say that the acceleration is so slow that almost all of friction is spent in providing the centripetal force in which it case it surely becomes level 3 ,

But either way, can you please show me how you solved the differential equation, to find velocity as function of angle

i believe that you have also done it in the same way by equating the resultant of friction to the net centripetal and tangential acceleration, but after that how you proceeded to solve it, Mvs Saketh · 2 years, 4 months ago

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@Mvs Saketh I'am Really Did not understand the Language , But Still I'am trying in this way ......

\(\displaystyle{mg\cos { \theta } -N=\cfrac { { mv }^{ 2 } }{ R } \quad (1)\\ ds=Rd\theta \quad (2)\\ \mu N-mg\sin { \theta } =mv\cfrac { dv }{ ds } \\ \\ \quad \quad \quad 2\mu N-2mg\sin { \theta } =m\cfrac { d({ v }^{ 2 }) }{ Rd\theta } \quad \quad \quad (3)}\).

from here we get V=f(\(\theta \)) and at V = max , accleration=0

Am I correctly understand This question If not , then What does This question Really Means ? Deepanshu Gupta · 2 years, 4 months ago

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@Deepanshu Gupta i dont know how to do this i found it on a book(physics today) but i think it is like what saketh said Visakh Radhakrishnan · 2 years, 4 months ago

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@Deepanshu Gupta Bro i believe you have assumed that he is travelling in a vertical circle, maybe the question means horizontal circle Mvs Saketh · 2 years, 4 months ago

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@Mvs Saketh it is horizontal Visakh Radhakrishnan · 2 years, 4 months ago

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@Mvs Saketh ohh :O , fish !

Lol I have Done unnecessary Calculations for vertical circle ! !\(\ddot\smile\)

But oK , if Now we create new question in which we consider vertical Circle , Then Is it correct ? Deepanshu Gupta · 2 years, 4 months ago

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@Deepanshu Gupta I Think so, It seems correct :) Mvs Saketh · 2 years, 4 months ago

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@Deepanshu Gupta sorry ,my bad.i forgot to tell you.I have edited it now Visakh Radhakrishnan · 2 years, 4 months ago

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