How can I translate a unit circle that doesn't start at 0, 1?

Question

Typically the unit circle "starts" at 0, 1. But lets say my vector starts at (x, y), what formula can I use to obtain the typical unit circle vectors $(\cos\theta, \sin\theta$)?

Answer 1

You may introduce a phase shift to start at whatever point that you like.

For example $$( x,y)= ( \cos(\theta + \pi/6), \sin (\theta + \pi/6) )$$ starts at $$(\cos(\pi/6), \sin ( \pi/6) )$$

Answer 2

We have that $(\cos \theta, \sin \theta)$ with $\theta\in[0,2\pi]$ describes a unit circle centered atbthe origin from $(1,0)$ counterclockwise.

To start from a generic point $(x,y)$ such that $x^2+y^2=1$ we need to take $\theta\in[\theta_0,\theta_0+2\pi]$ such that

• $x=\cos \theta_0$

• $y=\sin \theta_0$