How do you rewrite #cos 3theta# in terms of only #costheta# and #sintheta#?
1 Answer
Since
#\mathbf(sin(upmv) = sinucosv pm cosusinv)#
#\mathbf(cos(upmv) = cosucosv ∓ sinusinv)#
Thus:
#cos3theta#
#= cos(theta+2theta)#
#= costhetacolor(red)(cos2theta) - sinthetacolor(red)(sin2theta)#
Next, we still have
So, we have to rewrite
#color(green)(cos(theta+theta)) = costhetacostheta - sinthetasintheta#
#= color(green)(cos^2theta - sin^2theta)#
#color(green)(sin(theta+theta)) = sinthetacostheta + costhetasintheta#
#= color(green)(2sinthetacostheta)#
Thus, we end up with:
#color(blue)(cos3theta)#
#= costheta(color(green)(cos^2theta - sin^2theta)) - sintheta(color(green)(2sinthetacostheta))#
#= cos^3theta - sin^2thetacostheta - 2sin^2thetacostheta#
#= color(blue)(cos^3theta - 3sin^2thetacostheta)#