What is the Purpose of Phase Angle in Transformers?

It's the angle that the windings for each phase of the transformer gets pitched in wrapping around the core. You can measure this with a protractor by looking at the windings that touch the core and the lamination separations. The core is made with laminations to make it easier to measure this critical parameter to a transformer's efficiency.

And, if you have that twenty-two year old blonde technician make that measurement, you might even call it a cute angle.

open the cover of transformer. you can see the phase angle. No need any TTR instruments.Actually no purpose in designing, its just the winding machines design fault.

Its on the transformer name plate. E.g. DYN-11 etc...

You could using excitation current test to verify it.