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Question-154552




Question Number 154552 by 0731619 last updated on 19/Sep/21
Commented by Rasheed.Sindhi last updated on 20/Sep/21
•Is there any compitition to make  posts of others ′red-flaged′ ?  ♥Is there any compitition of ′likes′     of somebodies to their own posts?  ⧫ Is this a market place?  AND     winners are those who have more    ID′s or more divices!
$$\bullet\mathrm{Is}\:\mathrm{there}\:\mathrm{any}\:\mathrm{compitition}\:\mathrm{to}\:\mathrm{make} \\ $$$$\boldsymbol{\mathrm{posts}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{others}}\:'\mathrm{red}-\mathrm{flaged}'\:? \\ $$$$\heartsuit\mathrm{Is}\:\mathrm{there}\:\mathrm{any}\:\mathrm{compitition}\:\mathrm{of}\:'\mathrm{likes}' \\ $$$$\:\:\:\mathrm{of}\:\mathrm{somebodies}\:\mathrm{to}\:\boldsymbol{\mathrm{their}}\:\boldsymbol{\mathrm{own}}\:\boldsymbol{\mathrm{posts}}? \\ $$$$\blacklozenge\:\mathrm{Is}\:\mathrm{this}\:\mathrm{a}\:\boldsymbol{\mathrm{market}}\:\boldsymbol{\mathrm{place}}? \\ $$$$\boldsymbol{\mathrm{AND}}\: \\ $$$$\:\:\mathrm{winners}\:\mathrm{are}\:\mathrm{those}\:\mathrm{who}\:\mathrm{have}\:\mathrm{more} \\ $$$$\:\:\mathrm{ID}'\mathrm{s}\:\mathrm{or}\:\mathrm{more}\:\mathrm{divices}! \\ $$
Commented by Rasheed.Sindhi last updated on 19/Sep/21
@mathdanisur  How did you guess that?
$$@{mathdanisur} \\ $$$$\mathcal{H}{ow}\:{did}\:{you}\:{guess}\:{that}? \\ $$
Commented by Tinku Tara last updated on 20/Sep/21
Answering own question is very much  acceptable. Almost all senior mebers   of forum post a quesion which they  find interesting and answer in 2/3  days.  Posting very simple question just  increase noise and good challenging  questions end up getting ignored which  could create good discussion on solution  approaches as well.
$$\mathrm{Answering}\:\mathrm{own}\:\mathrm{question}\:\mathrm{is}\:\mathrm{very}\:\mathrm{much} \\ $$$$\mathrm{acceptable}.\:\mathrm{Almost}\:\mathrm{all}\:\mathrm{senior}\:\mathrm{mebers}\: \\ $$$$\mathrm{of}\:\mathrm{forum}\:\mathrm{post}\:\mathrm{a}\:\mathrm{quesion}\:\mathrm{which}\:\mathrm{they} \\ $$$$\mathrm{find}\:\mathrm{interesting}\:\mathrm{and}\:\mathrm{answer}\:\mathrm{in}\:\mathrm{2}/\mathrm{3} \\ $$$$\mathrm{days}. \\ $$$$\mathrm{Posting}\:\mathrm{very}\:\mathrm{simple}\:\mathrm{question}\:\mathrm{just} \\ $$$$\mathrm{increase}\:\mathrm{noise}\:\mathrm{and}\:\mathrm{good}\:\mathrm{challenging} \\ $$$$\mathrm{questions}\:\mathrm{end}\:\mathrm{up}\:\mathrm{getting}\:\mathrm{ignored}\:\mathrm{which} \\ $$$$\mathrm{could}\:\mathrm{create}\:\mathrm{good}\:\mathrm{discussion}\:\mathrm{on}\:\mathrm{solution} \\ $$$$\mathrm{approaches}\:\mathrm{as}\:\mathrm{well}. \\ $$
Commented by liberty last updated on 19/Sep/21
Commented by liberty last updated on 19/Sep/21
What are you saying ?
Commented by iloveisrael last updated on 20/Sep/21
Sieht aus als wärst du verrückt
Commented by mathdanisur last updated on 20/Sep/21
and  iloveisrael  as well as that person
$$\mathrm{and}\:\:\boldsymbol{\mathrm{i}}\mathrm{loveisrae}\boldsymbol{\mathrm{l}}\:\:\mathrm{as}\:\mathrm{well}\:\mathrm{as}\:\mathrm{that}\:\mathrm{person} \\ $$
Commented by iloveisrael last updated on 20/Sep/21
In my opinion, let it be. do we feel harmed by that? what's the downside? We don't need to interfere in all our people's affairs. if it doesn't hurt us
Commented by Rasheed.Sindhi last updated on 20/Sep/21
Then it will be difficult   for administator to distinguish                         between   “really able to be red marked”                              and   “marked red as a revenge”
$$\mathrm{Then}\:\mathrm{it}\:\mathrm{will}\:\mathrm{be}\:\mathrm{difficult}\: \\ $$$$\mathrm{for}\:\mathrm{administator}\:\mathrm{to}\:\mathrm{distinguish} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{between}\: \\ $$$$“{really}\:\boldsymbol{{able}}\:{to}\:{be}\:{red}\:{marked}'' \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{and}\: \\ $$$$“{marked}\:{red}\:{as}\:{a}\:\boldsymbol{{revenge}}'' \\ $$
Commented by mr W last updated on 20/Sep/21
forum is a public for exchanging  questions & answers in mathematics.  if somebody constantly posts pictures  or other things which have nothing  to do with exchanging mathematics,  i don′t feel hurt, but disturbed.
$${forum}\:{is}\:{a}\:{public}\:{for}\:{exchanging} \\ $$$${questions}\:\&\:{answers}\:{in}\:{mathematics}. \\ $$$${if}\:{somebody}\:{constantly}\:{posts}\:{pictures} \\ $$$${or}\:{other}\:{things}\:{which}\:{have}\:{nothing} \\ $$$${to}\:{do}\:{with}\:{exchanging}\:{mathematics}, \\ $$$${i}\:{don}'{t}\:{feel}\:{hurt},\:{but}\:{disturbed}. \\ $$
Commented by mathdanisur last updated on 20/Sep/21
Ser, there are 3 or more profiles of this  person and l mentioned it, he knowes  who he and that person paints what  l share in red, do his best, nothing will  chang!
$$\boldsymbol{\mathrm{S}}\mathrm{er},\:\mathrm{there}\:\mathrm{are}\:\mathrm{3}\:\mathrm{or}\:\mathrm{more}\:\mathrm{profiles}\:\mathrm{of}\:\mathrm{this} \\ $$$$\mathrm{person}\:\mathrm{and}\:\mathrm{l}\:\mathrm{mentioned}\:\mathrm{it},\:\mathrm{he}\:\mathrm{knowes} \\ $$$$\mathrm{who}\:\mathrm{he}\:\mathrm{and}\:\mathrm{that}\:\mathrm{person}\:\mathrm{paints}\:\mathrm{what} \\ $$$$\mathrm{l}\:\mathrm{share}\:\mathrm{in}\:\mathrm{red},\:\mathrm{do}\:\mathrm{his}\:\mathrm{best},\:\mathrm{nothing}\:\mathrm{will} \\ $$$$\mathrm{chang}! \\ $$
Commented by liberty last updated on 20/Sep/21
try to explain if someone posted a question and then someone answered whether it was the same person, what were your losses and what bothered you.
Commented by mr W last updated on 20/Sep/21
if someone don′t want to exchange   with others, he doesn′t need to use  the forum function. people who use  different identities to play the roles  of questioner and answerer as well  are cheating themselves.  i don′t  have any loss when other people   cheat themselves, i just think it′s  non−sense. or you don′t think so?
$${if}\:{someone}\:{don}'{t}\:{want}\:{to}\:{exchange}\: \\ $$$${with}\:{others},\:{he}\:{doesn}'{t}\:{need}\:{to}\:{use} \\ $$$${the}\:{forum}\:{function}.\:{people}\:{who}\:{use} \\ $$$${different}\:{identities}\:{to}\:{play}\:{the}\:{roles} \\ $$$${of}\:{questioner}\:{and}\:{answerer}\:{as}\:{well} \\ $$$${are}\:{cheating}\:{themselves}.\:\:{i}\:{don}'{t} \\ $$$${have}\:{any}\:{loss}\:{when}\:{other}\:{people}\: \\ $$$${cheat}\:{themselves},\:{i}\:{just}\:{think}\:{it}'{s} \\ $$$${non}−{sense}.\:{or}\:{you}\:{don}'{t}\:{think}\:{so}? \\ $$
Commented by mathdanisur last updated on 20/Sep/21
Ser, yes you are right, thanks.
$$\boldsymbol{\mathrm{S}}\mathrm{er},\:\mathrm{yes}\:\mathrm{you}\:\mathrm{are}\:\mathrm{right},\:\mathrm{thanks}. \\ $$
Commented by liberty last updated on 20/Sep/21
Mr. W, I noticed that you are the one who very often interferes in people's affairs, you position yourself as a traffic policeman who always determines whether people are right or wrong. those of you who always feel disturbed by posts like mister zainal. you are very great . 5 stars for you.
Commented by liberty last updated on 20/Sep/21
in this forum every member is free to post as long as what is posted does not offend or hurt us. let it be. After all, if you don't want to read it, don't read it. simple isn't it?
Commented by liberty last updated on 20/Sep/21
it's not a matter of reasonable or unreasonable. Don't make your measurements to measure others. the size we use for us personally. good for us is not necessarily good for others.
Commented by mr W last updated on 20/Sep/21
i don′t judge whether people are right  or wrong. i raraly red flag other  people′s posts. but when i see that  some people post things which are  not the original purpose of the forum,  i just tell them my opinion. since the   forum has no real administrator  in duty, it depents on us all if the  forum can remain clear. i personally  have nothing against anybody in the  forum.
$${i}\:{don}'{t}\:{judge}\:{whether}\:{people}\:{are}\:{right} \\ $$$${or}\:{wrong}.\:{i}\:{raraly}\:{red}\:{flag}\:{other} \\ $$$${people}'{s}\:{posts}.\:{but}\:{when}\:{i}\:{see}\:{that} \\ $$$${some}\:{people}\:{post}\:{things}\:{which}\:{are} \\ $$$${not}\:{the}\:{original}\:{purpose}\:{of}\:{the}\:{forum}, \\ $$$${i}\:{just}\:{tell}\:{them}\:{my}\:{opinion}.\:{since}\:{the}\: \\ $$$${forum}\:{has}\:{no}\:{real}\:{administrator} \\ $$$${in}\:{duty},\:{it}\:{depents}\:{on}\:{us}\:{all}\:{if}\:{the} \\ $$$${forum}\:{can}\:{remain}\:{clear}.\:{i}\:{personally} \\ $$$${have}\:{nothing}\:{against}\:{anybody}\:{in}\:{the} \\ $$$${forum}. \\ $$
Commented by mr W last updated on 20/Sep/21
if you let people do what has nothing  to do with the forum, the forum will  become a place full of carbage. do  you want it be such a place?
$${if}\:{you}\:{let}\:{people}\:{do}\:{what}\:{has}\:{nothing} \\ $$$${to}\:{do}\:{with}\:{the}\:{forum},\:{the}\:{forum}\:{will} \\ $$$${become}\:{a}\:{place}\:{full}\:{of}\:{carbage}.\:{do} \\ $$$${you}\:{want}\:{it}\:{be}\:{such}\:{a}\:{place}? \\ $$
Commented by liberty last updated on 20/Sep/21
Dear Mr. W, again I say if there are members who post questions and no one answers then those who post provide answers, leave it alone, I'm sure someone will benefit from that answer. on the forum there are many statuses whether it's student, student or teacher or lecturer. Again I say don't make your size to measure others. problem member using 2 ID leave it alone. Don't get distracted with other people's business. if you are nosy it just shows you are arrogant in this forum. 5 stars for you
Commented by talminator2856791 last updated on 20/Sep/21
 it is also annoying that someone    answers their own questions.
$$\:\mathrm{it}\:\mathrm{is}\:\mathrm{also}\:\mathrm{annoying}\:\mathrm{that}\:\mathrm{someone}\: \\ $$$$\:\mathrm{answers}\:\mathrm{their}\:\mathrm{own}\:\mathrm{questions}. \\ $$
Commented by liberty last updated on 20/Sep/21
that′s his problem , want to  post a question and he answers  himself let it be the problem.  if you feel annoying just keep  it in your heart that resentful  feeling. that′s your problem.
$${that}'{s}\:{his}\:{problem}\:,\:{want}\:{to} \\ $$$${post}\:{a}\:{question}\:{and}\:{he}\:{answers} \\ $$$${himself}\:{let}\:{it}\:{be}\:{the}\:{problem}. \\ $$$${if}\:{you}\:{feel}\:{annoying}\:{just}\:{keep} \\ $$$${it}\:{in}\:{your}\:{heart}\:{that}\:{resentful} \\ $$$${feeling}.\:{that}'{s}\:{your}\:{problem}. \\ $$
Commented by mr W last updated on 20/Sep/21
liberty sir: i decided not to discuss  with you further about this matter.  actually you didn′t understand and  din′t really look at what i did. i never  requested people to do things in the  way i like. i never deleted other  people′s posts because i don′t have  the power to do that even i want!  i even never thought that one may   not answer his own questions.  what i did is, when i see that somebody  doesn′t use the forum properly i try  to explain him what i think the  forum is for. sometimes i try to give  him concrete hints. all what i want  is that the forum is a clean place   where we like to be to exchange our  love to mathematics. certainly you  have the right to say that you don′t  care about some posts which some  other people don′t like. but you  should also accept that these other  people have the right not to like those  posts.   just image we have ten such persons  like that guy who constantly posts   his own home works in the forum   and ignores all kinds of communication  other people try with him. i think  after a few days the forum will  become a place full of carbage. i  don′t want this happen, therefore  i act.  it′s your good right to say that i were  arrogant. but i think the most people   here may judge by themselve who is  really arrogant.
$${liberty}\:{sir}:\:{i}\:{decided}\:{not}\:{to}\:{discuss} \\ $$$${with}\:{you}\:{further}\:{about}\:{this}\:{matter}. \\ $$$${actually}\:{you}\:{didn}'{t}\:{understand}\:{and} \\ $$$${din}'{t}\:{really}\:{look}\:{at}\:{what}\:{i}\:{did}.\:{i}\:{never} \\ $$$${requested}\:{people}\:{to}\:{do}\:{things}\:{in}\:{the} \\ $$$${way}\:{i}\:{like}.\:{i}\:{never}\:{deleted}\:{other} \\ $$$${people}'{s}\:{posts}\:{because}\:{i}\:{don}'{t}\:{have} \\ $$$${the}\:{power}\:{to}\:{do}\:{that}\:{even}\:{i}\:{want}! \\ $$$${i}\:{even}\:{never}\:{thought}\:{that}\:{one}\:{may}\: \\ $$$${not}\:{answer}\:{his}\:{own}\:{questions}. \\ $$$${what}\:{i}\:{did}\:{is},\:{when}\:{i}\:{see}\:{that}\:{somebody} \\ $$$${doesn}'{t}\:{use}\:{the}\:{forum}\:{properly}\:{i}\:{try} \\ $$$${to}\:{explain}\:{him}\:{what}\:{i}\:{think}\:{the} \\ $$$${forum}\:{is}\:{for}.\:{sometimes}\:{i}\:{try}\:{to}\:{give} \\ $$$${him}\:{concrete}\:{hints}.\:{all}\:{what}\:{i}\:{want} \\ $$$${is}\:{that}\:{the}\:{forum}\:{is}\:{a}\:{clean}\:{place}\: \\ $$$${where}\:{we}\:{like}\:{to}\:{be}\:{to}\:{exchange}\:{our} \\ $$$${love}\:{to}\:{mathematics}.\:{certainly}\:{you} \\ $$$${have}\:{the}\:{right}\:{to}\:{say}\:{that}\:{you}\:{don}'{t} \\ $$$${care}\:{about}\:{some}\:{posts}\:{which}\:{some} \\ $$$${other}\:{people}\:{don}'{t}\:{like}.\:{but}\:{you} \\ $$$${should}\:{also}\:{accept}\:{that}\:{these}\:{other} \\ $$$${people}\:{have}\:{the}\:{right}\:{not}\:{to}\:{like}\:{those} \\ $$$${posts}.\: \\ $$$${just}\:{image}\:{we}\:{have}\:{ten}\:{such}\:{persons} \\ $$$${like}\:{that}\:{guy}\:{who}\:{constantly}\:{posts}\: \\ $$$${his}\:{own}\:{home}\:{works}\:{in}\:{the}\:{forum}\: \\ $$$${and}\:{ignores}\:{all}\:{kinds}\:{of}\:{communication} \\ $$$${other}\:{people}\:{try}\:{with}\:{him}.\:{i}\:{think} \\ $$$${after}\:{a}\:{few}\:{days}\:{the}\:{forum}\:{will} \\ $$$${become}\:{a}\:{place}\:{full}\:{of}\:{carbage}.\:{i} \\ $$$${don}'{t}\:{want}\:{this}\:{happen},\:{therefore} \\ $$$${i}\:{act}. \\ $$$${it}'{s}\:{your}\:{good}\:{right}\:{to}\:{say}\:{that}\:{i}\:{were} \\ $$$${arrogant}.\:{but}\:{i}\:{think}\:{the}\:{most}\:{people}\: \\ $$$${here}\:{may}\:{judge}\:{by}\:{themselve}\:{who}\:{is} \\ $$$${really}\:{arrogant}. \\ $$
Commented by SLVR last updated on 26/Sep/21
  Mr..W...sir...afew^   people cannot  understand you +vely..  but i like that sacredness  thanks...to your services...  dear professor
$$ \\ $$$${Mr}..{W}…{sir}…{afe}\overset{} {{w}} \\ $$$${people}\:{cannot} \\ $$$${understand}\:{you}\:+{vely}.. \\ $$$${but}\:{i}\:{like}\:{that}\:{sacredness} \\ $$$${thanks}…{to}\:{your}\:{services}… \\ $$$${dear}\:{professor} \\ $$
Commented by mr W last updated on 26/Sep/21
thanks for understanding sir!
$${thanks}\:{for}\:{understanding}\:{sir}! \\ $$
Commented by Rasheed.Sindhi last updated on 26/Sep/21
Sir we know that you are not only  an expert mathematician but  also very sincere to the forum  and its members.
$$\mathcal{S}{ir}\:{we}\:{know}\:{that}\:{you}\:{are}\:{not}\:{only} \\ $$$${an}\:\boldsymbol{{expert}}\:\boldsymbol{{mathematician}}\:{but} \\ $$$${also}\:\boldsymbol{{very}}\:\boldsymbol{{sincere}}\:{to}\:{the}\:\boldsymbol{{forum}} \\ $$$${and}\:\boldsymbol{{its}}\:\boldsymbol{{members}}. \\ $$
Commented by mr W last updated on 26/Sep/21
thanks sir!  the same to you!
$${thanks}\:{sir}! \\ $$$${the}\:{same}\:{to}\:{you}! \\ $$
Commented by Rasheed.Sindhi last updated on 28/Sep/21
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Answered by MJS_new last updated on 19/Sep/21
squaring and transforming  x^(10) +x^8 −2x^6 −2x^4 +1=0  (x^4 +x^2 −1)(x^6 −x^2 −1)=0  x^2 =−(1/2)±((√5)/2) ∨x^2 =(((1/2)−((√(69))/(18))))^(1/3) +(((1/2)+((√(69))/(18))))^(1/3)   testing all solution I get for x∈R  x=−(√(−(1/2)+((√5)/2)))∨x=(√((((1/2)−((√(69))/(18))))^(1/3) +(((1/2)+((√(69))/(18))))^(1/3) ))
$$\mathrm{squaring}\:\mathrm{and}\:\mathrm{transforming} \\ $$$${x}^{\mathrm{10}} +{x}^{\mathrm{8}} −\mathrm{2}{x}^{\mathrm{6}} −\mathrm{2}{x}^{\mathrm{4}} +\mathrm{1}=\mathrm{0} \\ $$$$\left({x}^{\mathrm{4}} +{x}^{\mathrm{2}} −\mathrm{1}\right)\left({x}^{\mathrm{6}} −{x}^{\mathrm{2}} −\mathrm{1}\right)=\mathrm{0} \\ $$$${x}^{\mathrm{2}} =−\frac{\mathrm{1}}{\mathrm{2}}\pm\frac{\sqrt{\mathrm{5}}}{\mathrm{2}}\:\vee{x}^{\mathrm{2}} =\sqrt[{\mathrm{3}}]{\frac{\mathrm{1}}{\mathrm{2}}−\frac{\sqrt{\mathrm{69}}}{\mathrm{18}}}+\sqrt[{\mathrm{3}}]{\frac{\mathrm{1}}{\mathrm{2}}+\frac{\sqrt{\mathrm{69}}}{\mathrm{18}}} \\ $$$$\mathrm{testing}\:\mathrm{all}\:\mathrm{solution}\:\mathrm{I}\:\mathrm{get}\:\mathrm{for}\:{x}\in\mathbb{R} \\ $$$${x}=−\sqrt{−\frac{\mathrm{1}}{\mathrm{2}}+\frac{\sqrt{\mathrm{5}}}{\mathrm{2}}}\vee{x}=\sqrt{\sqrt[{\mathrm{3}}]{\frac{\mathrm{1}}{\mathrm{2}}−\frac{\sqrt{\mathrm{69}}}{\mathrm{18}}}+\sqrt[{\mathrm{3}}]{\frac{\mathrm{1}}{\mathrm{2}}+\frac{\sqrt{\mathrm{69}}}{\mathrm{18}}}} \\ $$
Answered by liberty last updated on 19/Sep/21
 x^3 −(1/x) = (1/( (√(x^2 +1))))   let x=tan θ   ⇒tan^3 θ−cot θ=(1/(sec θ))  ⇒tan^3 θ−cot θ=cos θ  ⇒((sin^3 θ)/(cos^3 θ))−((cos θ)/(sin θ))=cos θ  ⇒sin^4 θ−cos^4 θ=cos^4 θ sin θ  ⇒2cos^2 θ−1=cos^4 θ sin θ  ⇒2cos^2 θ−1=cos^4 θ (√(1−cos^2 θ))   ⇒2q−1=q^2  (√(1−q )) , q=cos^2 θ  ⇒4q^2 −4q+1=q^4 −q^5   ⇒q^5 −q^4 +4q^2 −4q+1=0  aproximation    { ((q≈0.430159=cos^2 θ)),((q≈0.618033=cos^2 θ)) :}
$$\:{x}^{\mathrm{3}} −\frac{\mathrm{1}}{{x}}\:=\:\frac{\mathrm{1}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}} \\ $$$$\:{let}\:{x}=\mathrm{tan}\:\theta\: \\ $$$$\Rightarrow\mathrm{tan}\:^{\mathrm{3}} \theta−\mathrm{cot}\:\theta=\frac{\mathrm{1}}{\mathrm{sec}\:\theta} \\ $$$$\Rightarrow\mathrm{tan}\:^{\mathrm{3}} \theta−\mathrm{cot}\:\theta=\mathrm{cos}\:\theta \\ $$$$\Rightarrow\frac{\mathrm{sin}\:^{\mathrm{3}} \theta}{\mathrm{cos}\:^{\mathrm{3}} \theta}−\frac{\mathrm{cos}\:\theta}{\mathrm{sin}\:\theta}=\mathrm{cos}\:\theta \\ $$$$\Rightarrow\mathrm{sin}\:^{\mathrm{4}} \theta−\mathrm{cos}\:^{\mathrm{4}} \theta=\mathrm{cos}\:^{\mathrm{4}} \theta\:\mathrm{sin}\:\theta \\ $$$$\Rightarrow\mathrm{2cos}\:^{\mathrm{2}} \theta−\mathrm{1}=\mathrm{cos}^{\mathrm{4}} \theta\:\mathrm{sin}\:\theta \\ $$$$\Rightarrow\mathrm{2cos}\:^{\mathrm{2}} \theta−\mathrm{1}=\mathrm{cos}\:^{\mathrm{4}} \theta\:\sqrt{\mathrm{1}−\mathrm{cos}\:^{\mathrm{2}} \theta}\: \\ $$$$\Rightarrow\mathrm{2}{q}−\mathrm{1}={q}^{\mathrm{2}} \:\sqrt{\mathrm{1}−{q}\:}\:,\:{q}=\mathrm{cos}\:^{\mathrm{2}} \theta \\ $$$$\Rightarrow\mathrm{4}{q}^{\mathrm{2}} −\mathrm{4}{q}+\mathrm{1}={q}^{\mathrm{4}} −{q}^{\mathrm{5}} \\ $$$$\Rightarrow{q}^{\mathrm{5}} −{q}^{\mathrm{4}} +\mathrm{4}{q}^{\mathrm{2}} −\mathrm{4}{q}+\mathrm{1}=\mathrm{0} \\ $$$${aproximation} \\ $$$$\:\begin{cases}{{q}\approx\mathrm{0}.\mathrm{430159}=\mathrm{cos}\:^{\mathrm{2}} \theta}\\{{q}\approx\mathrm{0}.\mathrm{618033}=\mathrm{cos}\:^{\mathrm{2}} \theta}\end{cases} \\ $$$$ \\ $$

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