The first part is easy- f(0)= 0- 3cos(0)= -1 and . Since f is negative at 0, positive at , and f is continuous, it must be 0 some place in between.
As for the second, do not know what the "Newton-Raphson" algorithm is? If your text has this question, surely it at least defines it?
The Newton-Raphson algorithm, to solve the equation f(x)= 0, requires that you define a sequence of x values, , where is some chosen value, hopefully near a root, and then .