How to solve for theta?

Question

cos(2 theta) = 4 - 7sin(theta) for 0 < or = to theta < or = to pi

Tried plugging some radians between that range, but can't get to a clear answer. :( Help?

Answer 1

Let Theta be A ( For simplicity ).

cos2A = 4-7sinA
=> 1-2(sinA)^2 = 4-7sinA
=> 2(sinA)^2-7sinA + 3 = 0
=> (sinA)^2 -(7/2)sinA + (3/2) = 0

=> (sinA)^2 -3sinA -(1/2)sinA + (3/2) = 0
=> sinA ( sinA -3) -(1/2) ( sinA - 3) = 0
=>(sinA - (1/2))(sinA - 3) = 0

Since, sinA lies between -1 and 1, it can't be equal to 3.
=> sinA - (1/2) = 0
=> sinA = 1/2
=>A = 30 Degrees
But since, 0<=A<=180
Answers: 30 Degrees and 150 Degrees

Answer 2

cos 2θ = 4 - 7 sin θ
1 - 2sin² θ = 4 - 7 sin θ
2 sin² θ - 7sinθ + 3 = 0
(2sinθ - 1).(sinθ - 3) = 0
sinθ = 1/2 is acceptable solution
θ = Π / 6 and 5Π / 6 (1st and 2nd quadrants)

Answer 3

cos 2A = 4 - 7 sin A
A = 30 & A = 150 deg