Through which angle must the axes be rotated so that the…

Through which angle must the axes be rotated so that the point $M(2,4)$ gets the coordinates $(4,-2)$ in the new system?

Explanation of Method

The rotation that gives M(2, 4) the new coordinates (4, −2) is a rotation by 90°.

Step 1: Rotation formulas

For axes rotated by $\varphi$: $x'=x\cos\varphi+y\sin\varphi,\qquad y'=-x\sin\varphi+y\cos\varphi$.

With $x=2,\ y=4,\ x'=4,\ y'=-2$: $2\cos\varphi+4\sin\varphi=4,\qquad -2\sin\varphi+4\cos\varphi=-2$.

Step 2: Solving

Multiply the first equation by $2$ and subtract the second: $10\sin\varphi=10$, so $\sin\varphi=1$. Then the first gives $\cos\varphi=0$.

\[\varphi=\frac{\pi}{2}=90^\circ\]

Check: $|OM|^2=4+16=20=16+4$ (a rotation preserves distance), and $x'=y=4$, $y'=-x=-2$.

$x'$ $y'$ $x'=4$ $y'=-2$ $\varphi=90^\circ$ $M$

Rotating the axes by $\varphi=90^\circ$: the new $x'$-axis is the old $y$-axis, the new $y'$-axis points along the old negative $x$-axis, so $M(2,4)$ becomes $M(4,-2)$.

Result

\[\boxed{\,\varphi=\frac{\pi}{2}=90^\circ\,}\]