Question

how many solution does this equation have between 0 and 4 pi? : cos(2x)=1/3.. not a fixed value in pi for arccos 1/3....

Answer 1

It has 8 solutions.
\begin{eqnarray*}
\cos 2x = 1/3
2x = \arccos 1/3 = 1.23
Using CAST Rule, there's another solution between 0 and 2pi at 2pi - 1.23 = 5.052.
Obviously we also have the following solutions too
2x = 1.23 + 2pi, 2x = 5.052 + 2pi, 2x = 1.23 + 4pi, 2x = 5.052 + 4pi, etc.
We then find x for each of these solutions until the solutions do not exceed 4pi, getting:
x=0.615,x=2.526, x=3.757, x=5.668, x=6.898, x=8.809, x=10.04, x=11.951.