Find all solutions of the equation sqrt(3)cos(x)=sin(x)+1 in the interval [0,2pi)

Question

Been stumped on this qusetion for a while, anyone got a simple step-by-step solution?

Answer 1

Square both sides

 

3cos²x = sin²x + 2 sin x + 1

 

Use the identity cos²x = 1 - sin²x

 

4 sin²x + 2 sin x - 2 = 0

2 sin²x + sin x - 1 = 0

(2 sin x - 1)(sin x + 1) = 0

 

sin x = 1/2 or sin x = -1

 

x = pi/6 = 30 degrees, x = 5pi/6 = 150 degrees or

x = 3pi/2 = 270 degrees