rewrite cos(x - 4pie/3) in terms of sin(x) and cos(x)

Answer 1

cos(x-[4∏/3])=cosxcos[4∏/3]+sinxsin[4∏/3]

=cosx[-1/2]+sinx[-sqrt(3)/2]

=-cosx/2-sinx[sqrt(3)/2]

=-cosx/2-sqrt(3)sinx/2

=(-cosx-sqrt(3)sinx)/2

=-(cosx+sqrt(3)sinx)/2

Answer 2

cos(x-y) = cosxcosy+sinxsiny (cos of the difference of two angles)

cos(x - 4pi/3) = cosx cos(4pi/3) + sinxsin(4pi/3)

cos(4pi/3) = -1/2

sin(4pi/3) = -√3/2

So:

(-1/2)cosx - (√3/2)sinx

check here for confirmation:

desmos.com/calculator/arurmkxy6u

Answer 3

Recall the cofunction identity cos(x-π/2) = sin(x)

cos(x-4π/3) = cos(x-π/2-5π/6) = sin(x-5π/6)

Notice how we write -4π/3 as -π/2-5π/6 so we can make that substitution.

Therefore, cos(x-4π/3) = sin(x-5π/6).

Hope this helped!