Rotation of Axes

When axes are rotated by angle θ anticlockwise, new coordinates (X, Y) relate to old (x, y) by these equations.

Let axes rotate anticlockwise by $\theta$, origin fixed. A point $P$ at distance $r$ from $O$ at angle $\phi$ in the old system has:

Old coordinates: $(r\cos\phi,\; r\sin\phi)$

New coordinates: $(r\cos(\phi-\theta),\; r\sin(\phi-\theta))$

Expanding:

$$X = r\cos(\phi-\theta) = r\cos\phi\cos\theta + r\sin\phi\sin\theta = x\cos\theta + y\sin\theta$$ $$Y = r\sin(\phi-\theta) = r\sin\phi\cos\theta - r\cos\phi\sin\theta = -x\sin\theta + y\cos\theta$$

The inverse (rotating back by $-\theta$):

$$\boxed{x = X\cos\theta - Y\sin\theta, \qquad y = X\sin\theta + Y\cos\theta}$$

What is preserved

Distances, angles, the origin, the degree of any equation.

Standard use

Choose $\theta$ to eliminate the $xy$ cross-term in a second degree equation. The required angle satisfies $\cot 2\theta = \dfrac{a-b}{2h}$. After rotation, the equation has the form $AX^2 + BY^2 + \ldots = 0$ — the conic type is immediately readable.