find all solutions of 2sinx=1-2cosx in the interval from 0 to 360

Answer 1

2sinx = 1 - 2cosx
sinx+cosx = 1/2

Now, sin(x+π/4) = sinx cosπ/4 + cosx sinπ/4
= 1/√2 (sinx+cosx)

SO, what you have is just

√2 sin(x+π/4) = 1/2
sin(x+π/4) = 1/√8
x+Ï€/4 = 0.368
x = -0.424

So, on [0,2Ï€) we have

x = 2Ï€-0.424 or (Ï€-0.368)-Ï€/4
x = 5.859 or 1.995