Solve for approximate solutions in the interval [0,2pi)

Question

2sin(θ)= 1- 2cos(θ)

Answer 1

The approximate solutions are:

1.99482736628563714

5.85915426768884595

 

which were found using the Bisection Method

Answer 2

Rewrite in terms of only sine or cosine.

 

Let x = θ

 

2√(1 - cos²x) = 1 - 2cosx

 

 

Square both sides of the equation.

 

4(1 - cos²x) = 1 - 4cosx + 4cos²x

 

4 - 4cos²x = 1 - 4cosx + 4cos²x

 

 

Move all terms to the right side of equation.

 

0 = 8cos²x - 4cosx - 3

 

 

Solve this quadratic equation for cosx.   Then use inverse cosine to find x.