Euler angle to cartesian angle conversion

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I need to calculate a new 2D point using a starting point and an Euler angle. But I'm having a hard time with converting the Euler angle to a cartesian angle.

I currently use Arturo Magidin's answer https://math.stackexchange.com/a/9367/554081 to calculate my new point using a starting point, the slope and a distance. Ideally, I want to insert the converted Euler angle into the current formula.

TLDR: How do I convert an Euler angle to a cartesian angle?