Expansion of cos 15 x interms of cos

[somani.utsav]

I have expanded cos 15x interms of cos. I hope some one who needs it can use it. I have verified this formulae using symbolic matlab: so its geniune

cos 15*theta = 16384*cos(theta)^15 - 61440*cos(theta)^13 + 92160*cos(theta)^11 - 70400*cos(theta)^9 + 28800*cos(theta)^7 - 6048*cos(theta)^5 + 560*cos(theta)^3 - 15*cos(theta)

 

[Mentallic]

Thanks, I've been looking everywhere for this!

 

[fbs7]

One might notice a similarity with the Chebyshev polynomial of 15th degree:

T15(x)=16384x¹⁵-61440x¹³+92160x¹¹-70400x⁹+28800x⁷-6048x⁵+560x³-15x

 

[somani.utsav]

This means that chebyshev function is nothing but a cos function

 

[AlephZero]

This means that chebyshev function is nothing but a cos function

That's not quite accurate, but $T_n(\cos \theta) = \cos(n \theta)$.

 

[D H]

This means that chebyshev function is nothing but a cos function

Does $16384 x^{15}-61440 x^{13}+92160 x^{11}-70400 x^9+28800 x^7-6048 x^5+560 x^3-15 x$ look like "a cos function"? It's a polynomial. Like the other Chebychev polynomials, it is a rather useful polynomial.

 

[somani.utsav]

That's not quite accurate, but $T_n(\cos \theta) = \cos(n \theta)$.

I retract my previous statement