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

Find the general solution of 2^(cos 2θ) +1=3.2^(−sin^2 θ)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{general}\:\mathrm{solution}\:\mathrm{of} \\ $$$$\mathrm{2}^{\mathrm{cos}\:\mathrm{2}\theta} +\mathrm{1}=\mathrm{3}.\mathrm{2}^{−\mathrm{sin}\:^{\mathrm{2}} \theta} \\ $$

Answered by Frix last updated on 20/Oct/22

2^(cos 2θ) +1=3×2^(−sin^2 θ) 2^(cos 2θ) +1=3×2^((cos 2θ−1)/2) 2^c +1=3×2^((c−1)/2) x^2 +1=(3/( (√2)))x x_1 =((√2)/2) ⇒ c_1 =−1 ⇒ cos 2θ =−1 x_2 =(√2) ⇒ c_2 =1 ⇒ cos 2θ =1 ⇒ θ=((nπ)/2)

$$\mathrm{2}^{\mathrm{cos}\:\mathrm{2}\theta} +\mathrm{1}=\mathrm{3}×\mathrm{2}^{−\mathrm{sin}^{\mathrm{2}} \:\theta} \\ $$$$\mathrm{2}^{\mathrm{cos}\:\mathrm{2}\theta} +\mathrm{1}=\mathrm{3}×\mathrm{2}^{\frac{\mathrm{cos}\:\mathrm{2}\theta−\mathrm{1}}{\mathrm{2}}} \\ $$$$\mathrm{2}^{{c}} +\mathrm{1}=\mathrm{3}×\mathrm{2}^{\frac{{c}−\mathrm{1}}{\mathrm{2}}} \\ $$$${x}^{\mathrm{2}} +\mathrm{1}=\frac{\mathrm{3}}{\:\sqrt{\mathrm{2}}}{x} \\ $$$${x}_{\mathrm{1}} =\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\:\Rightarrow\:{c}_{\mathrm{1}} =−\mathrm{1}\:\Rightarrow\:\mathrm{cos}\:\mathrm{2}\theta\:=−\mathrm{1} \\ $$$${x}_{\mathrm{2}} =\sqrt{\mathrm{2}}\:\Rightarrow\:{c}_{\mathrm{2}} =\mathrm{1}\:\Rightarrow\:\mathrm{cos}\:\mathrm{2}\theta\:=\mathrm{1} \\ $$$$\Rightarrow\:\theta=\frac{{n}\pi}{\mathrm{2}} \\ $$

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

Welldone sir MJS��������

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

Your sense of guessing is powerful. I remember that you had once guessed correctly for “Her-majesty” to be MJS sir! I think you′re also correct this time!

$${Your}\:{sense}\:{of}\:{guessing}\:{is}\:{powerful}. \\ $$$${I}\:{remember}\:{that}\:{you}\:{had}\:{once}\:{guessed} \\ $$$${correctly}\:{for}\:``{Her}-{majesty}''\:{to}\:{be} \\ $$$${MJS}\:\boldsymbol{{sir}}!\:{I}\:{think}\:{you}'{re}\:{also}\:{correct} \\ $$$${this}\:{time}! \\ $$

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

������

Commented by ARUNG_Brandon_MBU last updated on 21/Oct/22

He′s one of my favorite teachers and so I recognize his writings, his methods, expressions, and type of answers like finding the general term of a progression, haha. For example Q178467 Q177400. He loves letting t=x+(√(x^2 +1)) instead of using sinh in calculus. He often uses ∨ and ∧ to denote “or” and “and”. He likes saying “I can only approximate” then hits it hard with a tough solution. He also says “the rest is easy” when he′s busy to continue, “I get”. And lastly we′re very few in this forum who use Ostrogradsky and it was he who introduced it here as far as I know(I learnt it too from him). It wasn′t difficult. Haha ! I didn′t intend to call him MJS here. Just that he ignored my greetings. 😂😄😅

$$\:\:\:\:\:\:\:\:\mathrm{He}'\mathrm{s}\:\mathrm{one}\:\mathrm{of}\:\mathrm{my}\:\mathrm{favorite}\:\mathrm{teachers}\:\mathrm{and}\:\mathrm{so}\:\mathrm{I}\:\mathrm{recognize}\:\mathrm{his}\:\mathrm{writings},\:\mathrm{his}\:\mathrm{methods}, \\ $$$$\:\mathrm{expressions},\:\mathrm{and}\:\mathrm{type}\:\mathrm{of}\:\mathrm{answers}\:\mathrm{like}\:\mathrm{finding}\:\mathrm{the}\:\mathrm{general}\:\mathrm{term}\:\mathrm{of}\:\mathrm{a}\:\mathrm{progression}, \\ $$$$\:\mathrm{haha}.\:\mathrm{For}\:\mathrm{example}\:\mathrm{Q178467}\:\:\:\mathrm{Q177400}. \\ $$$$\:\mathrm{He}\:\mathrm{loves}\:\mathrm{letting}\:{t}={x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\:\mathrm{instead}\:\mathrm{of}\:\mathrm{using}\:\mathrm{sinh}\:\mathrm{in}\:\mathrm{calculus}.\:\mathrm{He}\:\mathrm{often}\:\mathrm{uses} \\ $$$$\vee\:\mathrm{and}\:\wedge\:\mathrm{to}\:\mathrm{denote}\:``\mathrm{or}''\:\mathrm{and}\:``\mathrm{and}''.\:\mathrm{He}\:\mathrm{likes}\:\mathrm{saying}\:``\mathrm{I}\:\mathrm{can}\:\mathrm{only}\:\mathrm{approximate}''\:\mathrm{then} \\ $$$$\mathrm{hits}\:\mathrm{it}\:\mathrm{hard}\:\mathrm{with}\:\mathrm{a}\:\mathrm{tough}\:\mathrm{solution}.\:\mathrm{He}\:\mathrm{also}\:\mathrm{says}\:``\mathrm{the}\:\mathrm{rest}\:\mathrm{is}\:\mathrm{easy}''\:\mathrm{when}\:\mathrm{he}'\mathrm{s}\:\mathrm{busy}\:\mathrm{to} \\ $$$$\mathrm{continue},\:``\mathrm{I}\:\mathrm{get}''.\:\mathrm{And}\:\mathrm{lastly}\:\mathrm{we}'\mathrm{re}\:\mathrm{very}\:\mathrm{few}\:\mathrm{in}\:\mathrm{this}\:\mathrm{forum}\:\mathrm{who}\:\mathrm{use}\:\mathrm{Ostrogradsky} \\ $$$$\mathrm{and}\:\mathrm{it}\:\mathrm{was}\:\mathrm{he}\:\mathrm{who}\:\mathrm{introduced}\:\mathrm{it}\:\mathrm{here}\:\mathrm{as}\:\mathrm{far}\:\mathrm{as}\:\mathrm{I}\:\mathrm{know}\left(\mathrm{I}\:\mathrm{learnt}\:\mathrm{it}\:\mathrm{too}\:\mathrm{from}\:\mathrm{him}\right).\: \\ $$$$\mathrm{It}\:\mathrm{wasn}'\mathrm{t}\:\mathrm{difficult}.\:\mathrm{Haha}\:!\:\mathrm{I}\:\mathrm{didn}'\mathrm{t}\:\mathrm{intend}\:\mathrm{to}\:\mathrm{call}\:\mathrm{him}\:\mathrm{MJS}\:\mathrm{here}.\:\mathrm{Just}\:\mathrm{that}\:\mathrm{he}\:\mathrm{ignored}\:\:\:\: \\ $$$$\mathrm{my}\:\mathrm{greetings}. \\ $$😂😄😅

Commented by ARUNG_Brandon_MBU last updated on 21/Oct/22

Q177400

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

You′re deep observer!!!

$${You}'{re}\:{deep}\:{observer}!!! \\ $$

Commented by ARUNG_Brandon_MBU last updated on 21/Oct/22

����

Commented by MJS_new last updated on 22/Oct/22

Mr. Sherlock Holmes please explain why I should have 2 accounts? Her Majesty had been my personal defense against the weirdness which had been creeping into this forum in those days... but now? it′s not my fault when someone uses the same language as me.

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

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(person)=account Certainly f is NOT afunction. So f(Person)=MJS,Frix,...is possible.I′m talking only about possibility.What is the probability that MJS=Frix ? This is an interesting question of probablity for forum members.I′m weak in this area but I think Mr Ar Brandon′s probability of correctness will be more than any other person (except MJS/Frix) in this connection :)

$${f}\left({person}\right)={account} \\ $$$${Certainly}\:{f}\:{is}\:{NOT}\:{afunction}. \\ $$$${So}\:{f}\left(\boldsymbol{\mathrm{Person}}\right)=\mathrm{MJS},\mathrm{Frix},...{is}\: \\ $$$${possible}.{I}'{m}\:{talking}\:{only}\:{about} \\ $$$$\boldsymbol{{possibility}}.{What}\:{is}\:{the}\:{probability} \\ $$$${that}\:\mathrm{MJS}=\mathrm{Frix}\:?\:\mathcal{T}{his}\:{is}\:{an}\:{interesting} \\ $$$${question}\:{of}\:{probablity}\:{for}\:{forum} \\ $$$${members}.{I}'{m}\:{weak}\:{in}\:{this} \\ $$$${area}\:{but}\:{I}\:{think}\:{Mr}\:{Ar}\:{Brandon}'{s} \\ $$$${probability}\:{of}\:{correctness}\:{will}\:{be} \\ $$$${more}\:{than}\:{any}\:{other}\:{person}\:\left({except}\right. \\ $$$$\left.{M}\left.{JS}/{Frix}\right)\:{in}\:{this}\:{connection}\::\right) \\ $$

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

��������

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

great sir Frix

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

Commented by Frix last updated on 24/Oct/22

thank you

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

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