Calculus, horizontal tangent

Question

For what values of x does the graph of f have a horizontal tangent? F(x)=x + 2sinx

Take the derivative and set it equal to zero, right? I got f'=1+2cosx and set it equal to zero and got x=2pi/3, but the book says (2n+1)pi + or - pi/3. Please help me make the connection, I have to show every step!

Answer 1

you are mostly right,

when does cos(x)=-1/2,

x₁=(2π/3)+2πk=(2π/3)(1+3k),,,,,,k=0,+/-1,+/-2,...

x₂=(-2π/3)+2πk=(2π/3)(3k-1),,,k=0,+/-1,+/-2,...

any of those values finds cos(x)=-1/2