It doesn't really change Euler's formula. It just adds a factor for x to it (or an additional term to hide inside sin and cos as functions). That's less conveni

It doesn't really change Euler's formula. It just adds a factor for x to it (or an additional term to hide inside sin and cos as functions). That's less convenient, not less true. However, I don't really see a benefit gained out of doing this to keep your theta between -1 and 1 instead of between -2pi and 2pi. Like it's not a difficult thing to estimate or convert in your head to get a close enough estimation.