At the high end of applied Trigonometry in Analytic Algebra or Precalculus courses we encounter the Rotation of Axes, always with conics. There is a practical use for this technology because it teaches us to interrelate quantities in different frames of reference.
The actual rotation requires significant calculations and the theory is developed by the trigonometric difference and multiple angle formulae. It's always a chore to do one of these rotations - but your students should be aware that they exist. A lot of simple yet elegant theory went into this discipline.
There are five Excel Spreadsheets here - Any one or all of them could be used as handouts although the fifth formula sheet is the only one designated as such.
We can generate a graph of this "tilted" conic without using rotation, but the resulting graph and its equation(s) are devoid of information. To handle a tilted conic in the familiar way, we must rotate the axes.
A Rotation of Axes by Orrin White is licensed under a Creative Commons Attribution 3.0 United States License