Rotating Coordinates: Solving for x and y

[Nusc]

If one rotates a tangent plane on a curved surface, this point can be expressed as follows:

x' = x cos(theta) - y sin(theta)
y' = x sin(theta) + y cos(theta)

One solves for x and y and computes based on the deviation of the deviation.

My question is: would the answer differ if you choose a different point say:

x' = x cos(theta) + y sin(theta)
y' = - x sin(theta) + y cos(theta)

note the negative sign.

 

[FactChecker]

Can you please explain your question some more? How could the answer NOT change if you change the equations and the resulting values?

 

[Nusc]

Can you please explain your question some more? How could the answer NOT change if you change the equations and the resulting values?

In the attached drawing, I can could approach this new point as

x' = x cos(theta) - y sin(theta)
y' = x sin(theta) + y cos(theta)

or

x' = x cos(theta) + y sin(theta)
y' = - x sin(theta) + y cos(theta)

From what you see in the diagram, how would you justify which coordinates?

 

[Nusc]

If you cannot, then what should the second pair of coordinates look like visually?

 

[FactChecker]

I still don't understand. In the diagram you posted, where are the points (x,y), (x',y'), and the angle theta?