So is the change in phase at zero crossing ?
No, phase changes when it needs to change. It does not affect anything. Things don't look good, but it does not matter for transmitters and receivers.
Plus, transmitters are rarely done this way, with a single multiplication. Typically you form baseband I/Q signal at some low frequency in a digital domain, pass it through the DAC, and then use quadrature mixer to bring it up to to the final RF frequency. This messes things up a lot towards the end anyway.
PS: And since the fist stage is digital, you can pick a baseband frequency that will work nicely.