How many angles \(\theta\in[0,2\pi]\) satisfy the equation \(\sin(\theta)+\sin(3\theta) = -\cos(\theta)+\cos(3\theta) ?\)
+128475 sin x + sin (3x) = -cos x + cos (3x)
sinx + sin2xcosx + sinxcos2x = - cosx + cos2xcosx - sin2xsin x
(sinx + cosx ) = - sin2x( six + cosx) + cos2x (cosx - sin x)
(sinx + cosx) = -2sinxcosx (sin x + cosx) + (cosxcosx - sinxsinx)(cosx - sinx)
(sin x + cosx) = -2sin^2xcosx - 2sinxcos^2x + cos^3x - sinxcos^2x - sin^2xcosx + sin^3x
sin x + cosx = = -3sin^2xcosx - 3cos^2x sinx +cos^3x + sin^3x
sin x + cosx = -3(1 - cos^2x)cosx - 3(1- sin^2x)sinx + cos^3x + sin^3x
sinx + cosx = -3cosx+ 3cos^3x - 3sinx + 3sin^3x + cos^3x + sin^3x
sin x + cos x = -3(sinx + cos x) + 4cos^3x + 4sin^3x
4 (sin x + cos x) = 4 (sin^3x + cos^3x) divide both sides by 4
sin x + cos x = sin^3x + cos^3x factor the right side as a sum of cubes
(sin x + cos x) = (sin x + cos x) (sin^2x - sinxcosx + cos^2x)
(sin x + cosx) = (sin x + cosx) ( 1 - sin xcosx)
(sin x + cosx) (1 - sin xcosx - 1) = 0
(sin x + cos x) ( -sin x cosx ) = 0
So
sin x + cosx = 0 and -sinx cosx = 0
x = 3pi/4 or 7pi/4 sinx cos x = 0
x = 0 or x = pi/2 or x = pi or x = 3pi/2 or x = 2pi
