Is there a formula like this???

[heo83]

Everybody knows that:
sin^2(x) = (1 - cos(2x))/2
cos^2(x) = (1 + cos(2x))/2
and
sin^3(x) = (3sin(x) - sin3(x))/4
cos^3(x) = (cos3(x) + 3cos(x))/4

Then, Is there a general formula for sin^n(x), cos^n(x) ????
Are you guys understand me? I want to eliminate all the exponent on the right hand side. Thanks if someone could answer my question. I really doubt whether this formula exists.
Thanks for reading

 

[Basil]

YES, the formulas you look for do exist. Nevertheless I bet you will not like it :twisted:

cos^n x=(1/2)^n-1 [cos nx + Cn,1 cos (n-2)x + Cn,2 cos (n-4)x + ... + Cn,(n-2)/2 cos 2x] + Cn,n/2 (1/2)^n

Cn,x = n!/[x! * (n-x)! ]


You also have available the "reverse" formula

cos nx = cos^n x - Cn,2 * cos^n-2 x * sin^2 x + Cn,4 * cos^n-4 x * sin^4 x - Cn,6 * cos^n-6 x * sin^6 x + ...


similar formulae for sin available (it's your home work :twisted: )

 

[heo83]

Maybe you want to know why I need the expanding formula for sin^n(x) and cos^n(x).
I have learnt Laplace Transform and there is an exercise:
L(sin^5(x)) = ?, is there a general formula for n ? the same for cos^n(x)
It would be easier for me to use Laplace Transform on the expanding formula.
Thanks for helping me.

 

[Jayson]

Here is the Laplace Transform for sin^5(x) is:

           120
 ----------------------
 (s^2+1)(s^2+9)(s^2+25)

sin^5(x) = -5/16sin(3x)+1/16sin(5x)+5/8sin(x)

for odd n: e.g sin^n(x) the Laplace Transform is:

               n!
 ---------------------------------------
 (s^2+1^2)(s^2+3^2)(s^2+5^2)...(s^2+n^2)

for even n: the Laplace Transform is:

               n!
 -------------------------------
 s(s^2+2^2)(s^2+4^2)...(s^2+n^2)

- Jayson

 

[heo83]

Jayson, how do you find out that formula?
Did you use Maple or Matlab or MathCad or Mathematica or something like that?
or you found them by yourself with a pen, a paper and a lot of thoughts? please show me!
Thanks

 

[waterman]

The attached file is my proof for cos^n(x). Any comment is welcome.
Sorry for using PDF format, because I can't get used to using the text format for exponentials.

 

[jpena]

You can find answers to formula generalizations in math handbooks, or math manuals (such as one written by a russian called Bronshtein).

 

[greek]

You can try Eli Maor's book - free from the publisher in PDF. You will learn lots.

 

[maharshi_qis]

ya..its there in all maths books..lik Gravel etc..