Rotation of Axes
Simple Explanation
Rotation of axes means rotating the coordinate axes by an angle θ about the same origin, without shifting it. A point's original coordinates relate to its coordinates in the rotated system by x = x'cosθ − y'sinθ and y = x'sinθ + y'cosθ.
Why Do We Need It?
Some curves (especially tilted conics) have much simpler equations once the axes are rotated to align with the curve's natural orientation — this technique makes that alignment possible.
See It
A sketch showing the original x and y axes as dashed lines, and a new set of axes x' and y' rotated by an angle theta from the originals, both through the same origin
Formula
Rotation of Axes
x = x'cosθ − y'sinθ, y = x'sinθ + y'cosθ
Relates a point's coordinates in an original (x,y) system to its coordinates in a new (x',y') system, rotated by angle θ about the same origin.
- θ
- — the angle the new axes are rotated by, relative to the old axes
When to use it: Whenever the coordinate axes need to be rotated to align with a tilted feature of a problem, simplifying its equation.
Worked Example
Convert coordinates to a rotated system
Axes are rotated by θ=90°. Find the new coordinates (x',y') of the point (5,0) in the old system.
Why Does This Work?
Rotating the axes by θ is mathematically equivalent to rotating every point by −θ relative to fixed axes — the rotation formulas come directly from the general definition of trigonometric ratios applied to this relative rotation.
Real-Life Example
Aligning coordinates with a tilted conveyor belt
An engineer analyzing motion along a conveyor belt that runs at an angle (not aligned with the standard horizontal/vertical directions) wants simpler equations of motion.
Rotating the coordinate axes to align with the belt's actual direction of travel turns an awkward tilted-motion problem into a simple straight-line one.
Practice
Axes are rotated by θ=90°. Find the new y'-coordinate of the point (0,4) in the old system.
HardCommon mistake
Using the θ-rotation formulas (for old coordinates in terms of new) when actually converting new from old requires the inverse (−θ) version — mixing up which direction the substitution goes.
Quick Review
- x = x'cosθ − y'sinθ, y = x'sinθ + y'cosθ.
- The inverse (solving for new from old) uses −θ in place of θ.
- Rotating axes can dramatically simplify a tilted curve's equation.