Solve 4 cos( θ )sin( θ ) − 2√3cos( θ ) − 2√3sin( θ ) + 3 = 0 . Give all the solutions in [0 , 360∘) .

Answer 1

4cosTsinT - 2sqr3(cosT) - 2sqr3(sinT) +3 =0

Theta = 30, 60, 120, 240 degrees or pi/6, pi/3, 2pi/3, 4pi/3

2(2cosTsinT) - 2sqr3(cosT +sinT) +3 = 0

sin2T -sqr3(cosT+sinT) =-3/2

sin2T +3/2 = sqr3(cosT+sinT)

sin^2(2T) + 3sin2T + 9/4 = 3(cos^2T + 2cosTsinT + sin^2T)= 3(1+sin2T)= 3 +3sin2T)

sin^2(2T) + 9/4 = 3

sin^2(2T) = 3/4

sin2T = sqr3/2

2T = 60, 120, 420, 480

T= 60/2, 120/2, 420/2, 480/2

T = 30, 60, 210, 240 or pi/6, pi/3, 7pi/6, 4pi/3,