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 + 2^{1/2}, 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°

Therefore,

x = x´ (2^{1/2}/2) - y´ (2^{1/2}/2)

y = x´ (2^{1/2}/2) + y´ (2^{1/2}/2)

Hence, the equation y = x + 2^{1/2} becomes:

x´ (2^{1/2}/2) + y´ (2^{1/2}/2) = x´ (2^{1/2}/2) - y´ (2^{1/2}/2) + 2^{1/2}

y´ = 1

In the new coordinates, the equation is a line parallel to the x´-axis, +1 unit away from the x´-axis.