Find-the-general-solution-of-2-cos-2-1-3-2-sin-2-

Question Number 178694 by peter frank last updated on 20/Oct/22

$$\mathrm{Find}\mathrm{the}\mathrm{general}\mathrm{solution}\mathrm{of}$$$$2^{\cos 2\theta }+1=3.2^{-\sin {}^2\theta }$$

Answered by Frix last updated on 20/Oct/22

$$2^{\cos 2\theta }+1=3\times 2^{-{\sin }^2\theta }$$$$2^{\cos 2\theta }+1=3\times 2^{\frac{\cos 2\theta -1}{2}}$$$$2^c+1=3\times 2^{\frac{c-1}{2}}$$$$x^2+1=\frac{3}{\sqrt{2}}x$$$$x_1=\frac{\sqrt{2}}{2}\Rightarrow c_1=-1\Rightarrow \cos 2\theta =-1$$$$x_2=\sqrt{2}\Rightarrow c_2=1\Rightarrow \cos 2\theta =1$$$$\Rightarrow \theta =\frac{n\pi }{2}$$

Commented by Ar Brandon last updated on 20/Oct/22

Welldone sir MJS��������

Commented by Rasheed.Sindhi last updated on 21/Oct/22

$$Yoursenseofguessingispowerful.$$$$Irememberthatyouhadonceguessed$$$$correctlyfor“Her-{majesty}^{″}tobe$$$$MJS\mathit{s}\mathit{i}\mathit{r}!Ithink{you}^{\prime }realsocorrect$$$$thistime!$$

Commented by Ar Brandon last updated on 21/Oct/22

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Commented by ARUNG_Brandon_MBU last updated on 21/Oct/22

$${\mathrm{He}}^{\prime }\mathrm{s}\mathrm{one}\mathrm{of}\mathrm{my}\mathrm{favorite}\mathrm{teachers}\mathrm{and}\mathrm{so}\mathrm{I}\mathrm{recognize}\mathrm{his}\mathrm{writings},\mathrm{his}\mathrm{methods},$$$$\mathrm{expressions},\mathrm{and}\mathrm{type}\mathrm{of}\mathrm{answers}\mathrm{like}\mathrm{finding}\mathrm{the}\mathrm{general}\mathrm{term}\mathrm{of}\mathrm{a}\mathrm{progression},$$$$\mathrm{haha}.\mathrm{For}\mathrm{example}\mathrm{Q}178467\mathrm{Q}177400.$$$$\mathrm{He}\mathrm{loves}\mathrm{letting}t=x+\sqrt{x^2+1}\mathrm{instead}\mathrm{of}\mathrm{using}\sinh \mathrm{in}\mathrm{calculus}.\mathrm{He}\mathrm{often}\mathrm{uses}$$$$\vee \mathrm{and}\wedge \mathrm{to}\mathrm{denote}“{\mathrm{or}}^{″}\mathrm{and}“{\mathrm{and}}^{″}.\mathrm{He}\mathrm{likes}\mathrm{saying}“\mathrm{I}\mathrm{can}\mathrm{only}{\mathrm{approximate}}^{″}\mathrm{then}$$$$\mathrm{hits}\mathrm{it}\mathrm{hard}\mathrm{with}\mathrm{a}\mathrm{tough}\mathrm{solution}.\mathrm{He}\mathrm{also}\mathrm{says}“\mathrm{the}\mathrm{rest}\mathrm{is}{\mathrm{easy}}^{″}\mathrm{when}{\mathrm{he}}^{\prime }\mathrm{s}\mathrm{busy}\mathrm{to}$$$$\mathrm{continue},“\mathrm{I}{\mathrm{get}}^{″}.\mathrm{And}\mathrm{lastly}{\mathrm{we}}^{\prime }\mathrm{re}\mathrm{very}\mathrm{few}\mathrm{in}\mathrm{this}\mathrm{forum}\mathrm{who}\mathrm{use}\mathrm{Ostrogradsky}$$$$\mathrm{and}\mathrm{it}\mathrm{was}\mathrm{he}\mathrm{who}\mathrm{introduced}\mathrm{it}\mathrm{here}\mathrm{as}\mathrm{far}\mathrm{as}\mathrm{I}\mathrm{know}\left(\mathrm{I}\mathrm{learnt}\mathrm{it}\mathrm{too}\mathrm{from}\mathrm{him}\right).$$$$\mathrm{It}{\mathrm{wasn}}^{\prime }\mathrm{t}\mathrm{difficult}.\mathrm{Haha}!\mathrm{I}{\mathrm{didn}}^{\prime }\mathrm{t}\mathrm{intend}\mathrm{to}\mathrm{call}\mathrm{him}\mathrm{MJS}\mathrm{here}.\mathrm{Just}\mathrm{that}\mathrm{he}\mathrm{ignored}$$$$\mathrm{my}\mathrm{greetings}.$$

Commented by ARUNG_Brandon_MBU last updated on 21/Oct/22

Q177400

Commented by Rasheed.Sindhi last updated on 21/Oct/22

$${You}^{\prime }redeepobserver!!!$$

Commented by ARUNG_Brandon_MBU last updated on 21/Oct/22

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Commented by MJS_new last updated on 22/Oct/22

$$\mathrm{Mr}.\mathrm{Sherlock}\mathrm{Holmes}\mathrm{please}\mathrm{explain}\mathrm{why}\mathrm{I}\mathrm{should}$$$$\mathrm{have}2\mathrm{accounts}?HerMajesty\mathrm{had}\mathrm{been}\mathrm{my}$$$$\mathrm{personal}\mathrm{defense}\mathrm{against}\mathrm{the}\mathrm{weirdness}\mathrm{which}$$$$\mathrm{had}\mathrm{been}\mathrm{creeping}\mathrm{into}\mathrm{this}\mathrm{forum}\mathrm{in}\mathrm{those}\mathrm{days}\dots$$$$\mathrm{but}\mathrm{now}?{\mathrm{it}}^{\prime }\mathrm{s}\mathrm{not}\mathrm{my}\mathrm{fault}\mathrm{when}\mathrm{someone}\mathrm{uses}$$$$\mathrm{the}\mathrm{same}\mathrm{language}\mathrm{as}\mathrm{me}.$$

Commented by ARUNG_Brandon_MBU last updated on 22/Oct/22

�� oh! "Her Majesty had been my personal defense against the weirdness which had been creeping into this forum in those days" But back then:- Her Majesty: "I have told you several times I am not MJS believe it or not." ���� You're funny, Sir. You made my day ��

Commented by Rasheed.Sindhi last updated on 22/Oct/22

$$f\left(person\right)=account$$$$CertainlyfisNOTafunction.$$$$Sof\left(\mathbf{Person}\right)=\mathrm{MJS},\mathrm{Frix},\dots is$$$$possible.I^{\prime }mtalkingonlyabout$$$$\mathit{p}\mathit{o}\mathit{s}\mathit{s}\mathit{i}\mathit{b}\mathit{i}\mathit{l}\mathit{i}\mathit{t}\mathit{y}.Whatistheprobability$$$$that\mathrm{MJS}=\mathrm{Frix}?\mathcal{T}hisisaninteresting$$$$questionofprobablityforforum$$$$members.I^{\prime }mweakinthis$$$$areabutIthinkMrAr{Brandon}^{\prime }s$$$$probabilityofcorrectnesswillbe$$$$morethananyotherperson(except$$$$MJS/Frix)inthisconnection:)$$

Commented by ARUNG_Brandon_MBU last updated on 22/Oct/22

���� Nice week-end to you, Sir RS ! Thanks for your company.

Commented by Rasheed.Sindhi last updated on 22/Oct/22

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Commented by peter frank last updated on 22/Oct/22

$$\mathrm{great}\mathrm{sir}\mathrm{Frix}$$

Commented by Frix last updated on 24/Oct/22

$$\mathrm{thank}\mathrm{you}$$

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