Changing the orientation of the reference axes while maintaining the origin. The main reason for rotating axes is that a given equation is much simpler in the new coordinate system than it was in the original system.
When the original x- and y-axes rotate counterclockwise through an angle , for any point P(x,y), the original coordinates (x,y) will become the new coordinates (x´,y´), and they are:
x´ = x cos + y sin
y´ = -x sin + y cos
To derive the equation in the new coordinates, we need to express original coordinates in the new coordinates:
x = x´ cos - y´ sin
y = x´ sin + y´ cos
For an example of rotation, consider a simple equation y = x + 21/2, which is a line. When the original x- and y-axes rotate counterclockwise through an angle of 45°, original coordinates can be expressed as:
x = x´ cos45° - y´ sin45°
y = x´ sin45° + y´ cos45°
x = x´ (21/2/2) - y´ (21/2/2)
y = x´ (21/2/2) + y´ (21/2/2)
Hence, the equation y = x + 21/2 becomes:
x´ (21/2/2) + y´ (21/2/2) = x´ (21/2/2) - y´ (21/2/2) + 21/2
y´ = 1
In the new coordinates, the equation is a line parallel to the x´-axis, +1 unit away from the x´-axis.