A “pattern” which breaks after n = 4. Any idea why?
I was experimenting with:
ƒ(x) = sin²ⁿ(x) + cos²ⁿ(x)
Where I found a pattern:
ƒ(x) = a⋅cos(4x) + (1-a) [a = (2ⁿ⁻¹-1)/2ⁿ]
The expression didn’t work at n = 0, but it seemed to hold for n = 1, 2, 3 and at n = 4 it finally broke. I don’t understand how from n = (1 to 3), ƒ(x) is a perfect sinusoidal wave but it fails to be one from after n = 4. Does anybody have any explanations as to why such pattern is followed and why does it break? (check out the attached desmos graph: https://www.desmos.com/calculator/p9boqzkvum )
As a side note, the expression: a⋅cos(4x) + (1-a), seems to be approaching: cos²(2x) as n→∞.
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@Apu
@Apu
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+solved @user1 @user2... to close the thread when your doubt is solved. Mention the users who helped you solve the doubt. This will be added to their stats.Note for OP
+solved @user1 @user2... to close the thread when your doubt is solved. Mention the users who helped you solve the doubt. This will be added to their stats.After n=4, both of the things are still sinusoidal though? Can you describe the "pattern" you mentioned?
Oh nvm
You mean the superposition for n=1,2,3 but not for other values of n.
Yes, f(x) for any other values is not sinusoidal.
That is true. Imma think about this :/
actually asked the same question on reddit, thoda sa complicated way mein samjhaya unhone, but can i send the link to the post here?
Yep.
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Oh, this explains it.
Imma copy paste the comment by u/chmath80:
f(x) = (cos²x)ⁿ + (1 - cos²x)ⁿ 2ⁿf(x) = (1 + cos2x)ⁿ + (1 - cos2x)ⁿ For n < 4, the rhs contains only constant terms, and multiples of cos²(2x), which equate to multiples of cos4x plus constants. For n > 3, there are also multiples of cos⁴(2x), which introduce multiples of cos8x, which breaks the pattern.@caffeine exploiter
haan i cant say that i truly understood everything, thoda bits and pieces mein hi samajh aaya
Yeah man. I gotta ponder over the complete thing too ig.
But one thing to note is patterns involving 1/2^n can be misleading
When figured out by intuition.
i see
nahi i mean the “a” constant which i figured out was purely from pattern recognition
it does not even play a big role as to why for n > 3, f(x) does not remain sinusoidal
even removing the constant, the cos(4x) expression is obviously sinusoidal, but f(x) isn’t for n>3
although i do see your point.