I'm confused, in part a they said draw the frequency domain equivalent circuit shouldn't we write the current source as a complex number ?

For me frequency domain = phasors = complext numbers

In part b they ask find the phasor voltage but they give the answer in terms of theta. What am I missing ?

 

The top line of the written solution should be Io = ... (the current source phasor) rather than Vo = .... It's just a typo error.

The voltage Vo is then found by calculating the equivalent impedance and multiplying by Io - which is all the question was asking in a somewhat convoluted manner.

 

I'm confused, in part a they said draw the frequency domain equivalent circuit shouldn't we write the current source as a complex number ?

For me frequency domain = phasors = complext numbers

In the frequency domain circuit representation the current source is denoted by the polar form of the complex number - which is equivalent to the rectangular form via the standard polar to rectangular conversion. There's no ambiguity, whether you represent in either rectangular or polar. Both are equally valid. One normally knows what is implied in the context of the notation.

 

In part b they ask find the phasor voltage but they give the answer in terms of theta. What am I missing ?

A complete solution for the required voltage as a phasor should include both magnitude and phase angle - so why would you not include the phase angle [theta]? When you construct a phasor diagram don't you draw it with consideration to both individual phasor magnitudes and the relative phase displacements of the various phasor quantities shown in the diagram?