sin(x/2)+cos(x)=0

Question

I believe that I'm supposed to use a double angle identity but I don't know exactly how to work the problem. 

Answer 1

Let us set

θ = x/2

 

sin θ + cos 2θ = 0

sin θ + cos² θ - sin² θ = 0          cos² = 1 - sin²

sin θ + 1 - sin² θ - sin² θ = 0

-2sin² θ + sin θ + 1 = 0

2sin² θ - sin θ - 1 = 0

(2sin θ + 1)(sin θ - 1) = 0

 

2sin θ = -1

sin θ = -1/2

θ ∈ {7π/6, 11π/6}, x ∈ {7π/3, 11π/3}

 

sin θ = 1

θ ∈ {π/2}, x ∈ {π}