Equivalent expression for sinx^3

[synergix]

Homework Statement

Find an equivalent expression for sin³x in terms of function values of x, 2x, 3x, raised only to the first power.

The Attempt at a Solution

stuck! don't know where to begin I am reviewing for my exam right now I remember doing one of these problems before but I can't remember how.

sinx(sinx²)

sinx[(1-cos2x)/2]

am i on the right track?

 

[tiny-tim]

Find an equivalent expression for sin³x in terms of function values of x, 2x, 3x, raised only to the first power.
…
stuck! don't know where to begin I am reviewing for my exam right now I remember doing one of these problems before but I can't remember how.

sinx(sinx²)

sinx[(1-cos2x)/2]

am i on the right track?

Hi synergix!

(btw, don't write (sinx²), write either (sinx)² or sin²x)

Your equation is correct, but I don't think it's what they're asking for …

it has a (sinx)(cos2x) … if you're allowed that, you might as well have (sinx)(sinx)(sinx).

I think they want a sum of first powers, each on its own.

Hint: start at the end …

what is sin(3x) in terms of powers of sinx?

 

[Architeuthis Dux]

Yes, you are on the right track. Here is one possible solution:

sin3x = sin(2x + x) = sin2x*cosx + cos2x*sinx

= (2sinx*cosx)*cosx + (cos^2x - sin^2x)*sinx

= 2sinx*cos^2x + cos^2x*sinx - sin^3x

= sinx(2cos^2x + cos^2x - sin^2x)

= sinx(3cos^2x - sin^2x)

= sinx(3(1-sin^2x) - sin^2x) [using the identity cos^2x = 1-sin^2x]

= sinx(3-3sin^2x - sin^2x)

= sinx(3-4sin^2x)

Therefore, an equivalent expression for sin3x in terms of function values of x, 2x, 3x, raised only to the first power is 3sinx - 4sin^3x.