What is #cos^3theta-cos^2theta+costheta# in terms of non-exponential trigonometric functions?

2 Answers
Bio
Feb 17, 2016

#cos^3 theta - cos^2 theta + cos theta = frac{cos 3theta}{4} - frac{cos 2theta}{2} + frac{7 cos theta}{4} - 1/2#

Explanation:

Do you know the compound angle identities for #cos theta#?

#cos 2theta -= 2cos^2 theta -1#

#cos 3theta -= 4cos^3 theta - 3cos theta#

Change the subject of the formula.

#cos^2 theta -= frac{cos 2theta + 1}{2}#

#cos^3 theta -= frac{cos 3theta + 3cos theta}{4}#

So substitute them into your expression.

#cos^3 theta - cos^2 theta + cos theta #

#= frac{cos 3theta + 3cos theta}{4} - frac{cos 2theta + 1}{2} + cos theta#

#= frac{cos 3theta}{4} - frac{cos 2theta}{2} + frac{7 cos theta}{4} - 1/2#

Nghi N.
Feb 17, 2016

#f(x) = cos x((1 + cos 2x)/2 - cos x + 1)#

Explanation:

Put cos x in common factor -->
#f(x) = cos x(cos^2 x - cos x + 1)#.
Replace #cos^2 x# by #((1 + cos 2x)/2)#
#f(x) = cos x[(1 - cos 2x)/2 - cos x + 1] =#

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