Basically, write \cos 4x as a polynomial in \sin x. I've

Kaspaueru2

Kaspaueru2

Answered question

2021-12-31

Basically, write cos4x as a polynomial in sinx.
I've tried the double angles theorem and cos2x=cos2xsin2x. I'm still having trouble right now though.

Answer & Explanation

rodclassique4r

rodclassique4r

Beginner2022-01-01Added 37 answers

cos4x=12sin22x
=12(1cos22x)
=12(1(12sin2x)2)
=12(4sin2x4sin4x)
=18sin2x+8sin4x
scomparve5j

scomparve5j

Beginner2022-01-02Added 38 answers

If you prefer we can use complex numbers
Let z=cosx+isinx. Using de Moivres
Vasquez

Vasquez

Expert2022-01-08Added 669 answers

Double angle theorem once:
cos4x=cos22x+sin22x
Double angle theorem twice:
=(cos2xsin2x)2+(2sinxcosx)2)
Expand chicken soup and rice:
=cos4x2cos2xsin2x+sin4x+4sin2xcos2x)
To get it entirely in terms of sinkx replace cos2x with 1sin2x (keeping in mind cos4x=(cos2x)2)
=(1sin2x)22(1sin2x)sin2x+sin4x+4sin2x(1sin2x)

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