Thanks a lot! I am also so glad to be back. I have been booked by some projects, but from now on I will be posting more videos. Next few, however, will be more about the cubic equations. I would like to present the full details of the topic.
My calculator's Computer Algebra System (CAS) doesn't have an exact expression for pi/9 radians, which means it doesn't have an exact expression for cos(pi/9). So I don't know how to substitute your solution into the original equation to prove it's correct, except as an approximation.
Thank you very much for your kind comment. Soon I will post a series of videos about the details of the cubic equations and we will see different ways of solving cubic equations.
cos(3x)=4cos^3(x)-3cos(x) This remids me Chebyshov polynomials (I am not expert in Russian but you read this name wrong) I tried to get formula for these polynomials but when i went from T_{n}(x) = cos(narccos(x)) to the recurrence relation i solved with exponential generating function , Laplace transform, Leibniz product rule , binomial expansion and stopped on double sum (because I dont know how to finish it) I tried to get formula for this polynomials but when i went from T_{n}(x) = cos(narccos(x)) to the ordinary differential equation i was unable to get initial conditions So I couldnt calculate a_{0} and a_{1} in series solution of that ode
It's great to see you back! Thank you for another excellent video.
Thanks a lot! I am also so glad to be back. I have been booked by some projects, but from now on I will be posting more videos. Next few, however, will be more about the cubic equations. I would like to present the full details of the topic.
Very interesting, thank you.
You're very welcome! Soon I will post more videos on how to solve cubic equations.
This is an amazing and very informative video on cubic equations. Thank you!
My calculator's Computer Algebra System (CAS) doesn't have an exact expression for pi/9 radians, which means it doesn't have an exact expression for cos(pi/9). So I don't know how to substitute your solution into the original equation to prove it's correct, except as an approximation.
Nice problem and solution :)
Thank you very much for your kind comment. Soon I will post a series of videos about the details of the cubic equations and we will see different ways of solving cubic equations.
cos(3x)=4cos^3(x)-3cos(x)
This remids me Chebyshov polynomials
(I am not expert in Russian but you read this name wrong)
I tried to get formula for these polynomials but when i went from
T_{n}(x) = cos(narccos(x))
to the recurrence relation
i solved with exponential generating function , Laplace transform, Leibniz product rule , binomial expansion
and stopped on double sum (because I dont know how to finish it)
I tried to get formula for this polynomials but when i went from
T_{n}(x) = cos(narccos(x))
to the ordinary differential equation i was unable to get initial conditions
So I couldnt calculate a_{0} and a_{1} in series solution of that ode