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\( \\ Given\quad \sin { \theta } +\sin ^{ 2 }{ \theta } =1\\ find\quad \cos ^{ 12 }{ \theta } +3\cos ^{ 10 }{ \theta } +3\cos ^{ 8 }{ \theta } +\cos ^{ 6 }{ \theta } \quad -1\quad .\)

This was my first question in brilliant ( excluding the question: next number in the sequence 2,4,8.. during account creation).

\( Since\quad value\quad is\quad given\quad in\quad terms\quad of\quad sin\theta \quad I\quad thought\\ it\quad might\quad be\quad beneficial\quad to\quad convert\quad terms\quad of\quad cos\theta \quad in\quad \\ sin\theta \quad moreover\quad its\quad apparent\quad that\quad it\quad can\quad be\quad done\quad easily.\quad \\ After\quad substituting(\cos ^{ 2 }{ \theta } =\sin { \theta } )\quad I\quad got\quad these\quad terms:\\ \\ \sin ^{ 3 }{ \theta } +\sin ^{ 6 }{ \theta } +3\sin ^{ 4 }{ \theta } +3\sin ^{ 5 }{ \theta } -1\\ \\ Which\quad on\quad rearranging\quad gave\quad these:\\ \\ \Rightarrow \sin ^{ 3 }{ \theta } +\sin ^{ 6 }{ \theta } +3\sin ^{ 3 }{ \theta } (\sin ^{ 2 }{ \theta } +\sin { \theta } )-1\\ \\ I\quad finally\quad shot\quad the\quad problem\quad by\quad this:\\ \Rightarrow { (\sin { \theta } +\sin ^{ 2 }{ \theta } ) }^{ 3 }-1=0.\\ \\ I\quad used\quad to\quad face\quad (to\quad be\quad honest,I\quad still\quad face)\quad difficulty\quad while\quad \\ attempting\quad 'trigo'\quad problems.They\quad appear\quad to\quad me\quad so\quad absurd.\\ 'What\quad should\quad we\quad do\quad here?'\\ 'Why\quad would\quad anyone\quad do\quad that?'\\ 'OK!\quad good\quad solution\quad there\quad but\quad how\quad to\quad arrive\quad at\quad it?'\\ These\quad are\quad the\quad family\quad of\quad questions\quad clouding\quad over\quad my\\ mind.\quad So,\quad I\quad (honestly\quad try)\quad to\quad reason\quad out\quad my\quad steps.\\ \\ I\quad am\quad not\quad good\quad at\quad Mathematics.\quad But,\quad I\quad like\quad Maths.\quad \\ Here\quad in\quad brilliant.org,\quad there\quad are\quad very\quad talented\quad problem\quad solvers.\\ I\quad know\quad it\quad because\quad I\quad had\quad spent\quad some\quad time\quad with\quad my\quad friend,\quad \\ in\quad his\quad profile,\quad tackling\quad problems\quad and\quad viewing\\ solutions\quad and\quad reading\quad notes.\quad Then\quad it\quad allured\quad me\quad to\quad join\quad it.\\ \\ I\quad don't\quad have\quad collections\quad of\quad nut-cracking\quad problems\quad neither\quad \\ the\quad ability\quad to\quad shoot\quad those\quad type\quad of\quad questions\quad but\quad still\quad if\\ some\quad are\quad benefited\\ in\quad any\quad kind\quad through\quad me\quad \\ \\ I\quad would\quad be\quad glad.\\ \\ \)

Note by Soumo Mukherjee
2 years ago

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Wassup Math Philic!

I'm John - and I post the strangest problems around here ;)

I remember the first time I've discovered Brilliant. I Googled "very hard math problems," and clicked a link that led me here - and tell ya what, it's been one hella ride. Throughout the time being, I've discovered a community of intelligent and respectful individuals around the world who not only share math and physics problems and solutions but are willing to address your problems when you need it the most.

Let me introduce you to this site's Bomb Squad:

@Calvin Lin - The White Knight (if he reshares your problem, it's your lucky day!)

@Michael Mendrin - Mr. Mathopedia

@hasan kassim - King Integral

@Mardokay Mosazghi - coolest kid in town.

@jatin yadav - "can't" ain't in his dictionary.

@brian charlesworth - Dr. Philosophy

@Marios Patsios - don't mess with this one...

@David Mattingly - PHYSICS 101

@Arron Kau - Updates!

@Peter Taylor - look at those mountains...


@Suyeon Khim - the one who started it all.



There's many more, sorry if I forgot anyone, but trust me, there's WAY more...

Oh yeah...

And hey, don't be shy to try one of my most overrated problems about the Moon and the Universe: How to Climb to the Moon - it's a real teaser!

Oh, and we don't just have problems; we can discuss things all we want. Try out Physics Jokes!

Welcome to the club.



John Muradeli · 2 years ago

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@John Muradeli Haha... "Dr. Philosophy" ... You're too funny. :) There were a couple of people on the "Bomb Squad" I wasn't familiar with, so thanks for making the list. I would add @Jon Haussmann as well as "The Annihilator" for the way he mows down countless difficult problems like they're nothin'.

@Math Philic Welcome! Just thought I'd offer a couple of pieces of advice:

  • If you want more problems, notes, etc., to appear on your feed, just follow those you admire and then their posts and reshares will be placed on your feed.

  • I notice that all your text is italicized. Most people find this a bit more difficult to read than standard font, so you may want to just use the LaTeX for the "math parts" of your posts. :)

Brian Charlesworth · 2 years ago

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@Brian Charlesworth yes, for the use of latex part one of the moderator @Aditya Raut has helped me. Thanx to u 2 guys.I gonna follow the mentioned guys.. :) Soumo Mukherjee · 2 years ago

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@Brian Charlesworth 1 more thing.How to mention a name in comment? Soumo Mukherjee · 2 years ago

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@Soumo Mukherjee To mention a name, type the @ symbol followed by the name you wish to mention. Usually, as you're typing out the letters of the name there will be a drop-down list of names that match the letters typed to that point from which you can pick, (this option saves a bit of time). The person will be notified in their feed that they have been mentioned. Brian Charlesworth · 2 years ago

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@John Muradeli U missed urself-the cool guy.. thanx mate for this intro.. Soumo Mukherjee · 2 years ago

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