Recall that #sin(3x) = sin(2x+ x)#.
We use the sum formula #sin(A + B) = sinAcosB + sinBcosA# to expand #sin(2x + x)#.
#sin2xcosx + cos2xsinx - sinx = 1#
#2sinxcosx(cosx) +( 2cos^2x - 1)sinx - sinx = 1#
#2sinxcos^2x + 2cos^2xsinx - sinx - sinx = 1#
#4sinxcos^2x - 2sinx = 1#
#2sinx(2cos^2x- 1) = 1#
#2cos^2x - 1 = 1/(2sinx)#
#2cos^2x = 1+ 1/(2sinx)#
#2(1 - sin^2x) = (2sinx + 1)/(2sinx)#
#2 (1 - sin^2x) = (2(sinx + 1/2))/(2sinx)#
#2(1 - sin^2x) = (sinx + 1/2)/(sinx)#
#(2 - 2sin^2x )sinx = sinx + 1/2#
#2sinx - 2sin^3x = sinx + 1/2#
#-2sin^3x + sinx - 1/2 = 0#
We let #t= sinx#.
#-2t^3 + t - 1/2 = 0#
Solve using a graphing calculator, to get #t ~= -0.885#.
#sinx= -0.885#
#x = pi + 2pin+ arcsin(0.885)# and #2pin - arcsin(0.885)#
Hopefully this helps!
