Excuse my poor description in the title, I think a picture is needed to explain my question:
Theta is the angle to the x-axis.
So my question is: given the radius of the circle, theta, and beta, what is theta'?
The problem arises when simulating a car (with heading theta) that is turning. Beta can be assumed to be small.
Where I saw this problem it was stated: theta' = theta+beta % 2*pi. But I want to know why.
This is about the rotation of the tangent line and horizontal line by $\beta$.
If you rotate the first tangent line by $\beta$ you get the second tangent line.
Then the first horizontal line is rotated by $\beta$ and still makes the angle $\theta$ with the rotated tangent line.
However the angle between the rotated horizontal line and the new horizontal line is $\beta$ and hence the whole angle is $\theta+\beta$.