How do I calculate the new coordinates after rotating the coordinate axes?

In summary, when rotating a point (x,y) by an angle θ, the new coordinates (x',y') can be found using the formulas x' = xcosθ + ysinθ and y' = -xsinθ + ycosθ. This can also be derived by using angle addition formulas for cosine and sine and substituting the polar coordinates (R, phi) for the point. To convert these formulas for clockwise rotations, substitute -theta in place of theta.
  • #1
CrazyNeutrino
100
0
If I have a point (x,y) and I rotate the axises by some amount. Why is x' = xcosθ+ysinθ and y'=-xsinθ+ycosθ?
 
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  • #2
Look at it this way. Suppose (x,y) has polar coords R, phi. Then the rotated point has polar coords R, phi + theta. So the new rectangular coords should be

x' = Rcos(phi+theta)
y'= Rsin(phi+theta)

Now use the angle addition formulas for cosine and sine and use the fact that
x=Rcos(phi)
y=Rsin(phi)
 
  • #3
Angle addition formulas for what?
I don't understand you fully :(
 
  • #4
I understand that x' = rcos(theta+phi) and y'= rsin(theta+phi) and that x=rcos theta and y= rsin theta.
How do I use this to get x'= xcos theta + ysin theta
And y= -xsin theta + ycos theta
 
  • #5
The last sentence is actually y'. Sorry
 
  • #6
cos (α + β) = cos α cos β − sin α sin β

sin (α + β) = sin α cos β + cos α sin β

Using this, I can write, x'=rcos theta cos phi - sin theta sin phi
and y'= rsin theta cos phi + cos theta sin phi

and x= rcos phi
y= rsin phi

Now what do I do?
 
  • #7
Now substitute x=rcos(phi) and y=rsin(phi) into those expressions.

You will get formulas that are almost the same as what you started with. The difference is due to the following:

The formula you derived answers the following. Given a counter clockwise rotation of the point, what are its new coordinates.

Your original question was this: If we rotate the coordinate axes, what are the new coordinates with respect to the rotated axes. To answer this we have to realize that rotating the coordinate axes counterclockwise is equivalent to rotating the points clockwise. So your new coordinates will be the coordinates you get after rotating your point clockwise. But the formula you derived is valid for counterclockwise rotations. To convert it, you must substitute -theta in place of theta. Then you will get the formula that you first asked about.
 

1. What is a co-ordinate plane rotation?

A co-ordinate plane rotation is a geometric transformation in which a point or shape on a coordinate plane is rotated by a certain degree around a fixed point, also known as the center of rotation.

2. How is a co-ordinate plane rotation measured?

A co-ordinate plane rotation is measured in degrees, with positive values indicating a clockwise rotation and negative values indicating a counterclockwise rotation.

3. What is the purpose of a co-ordinate plane rotation?

The purpose of a co-ordinate plane rotation is to change the position or orientation of a point or shape on a coordinate plane without changing its size or shape. It is commonly used in geometry and mathematics to solve problems involving angles and positions of objects.

4. How do you perform a co-ordinate plane rotation?

To perform a co-ordinate plane rotation, you must first identify the center of rotation and the angle of rotation. Then, you can use a rotation formula or rule to calculate the new coordinates of the point or shape after rotation.

5. Can a co-ordinate plane rotation be applied to three-dimensional objects?

Yes, a co-ordinate plane rotation can be applied to three-dimensional objects by rotating them around an axis. This is commonly used in computer graphics and 3D modeling.

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