Rotation of Axis, Concept, and Examples:

Rotation of Axis: Let $xy$ -coordinate system be given. We rotate $Ox$ and $Oy$ and about the origin through an angle $\theta \left(0<\theta<90^\circ\right)$ so that the new axes are $OX$ and $OY$ as shown in the figure. Let a point $P$ have coordinates $\left(x,y\right)$ referred to the $xy$ -system of coordinates. Suppose $P$ has coordinates $(X, Y)$ referred to the $XY$ -coordinate system. We have to find $X, Y$ in terms of the given coordinates $x, y$ . Let a be measure of the angle that $OP$ makes with $O$

From $P$ , draw $PM$ perpendicular to $Ox$ and PM’ perpendicular to $OX$ . Let $\left|OP\right|=r$ ,

From the right triangle $OPM'$ , we have

$OM'=X=r\;\cos\left(\alpha-\theta\right)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (1)$ .

and

$M'P=Y=r\;\sin\left(\alpha-\theta\right)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(2)$

Also from triangle $OPM$ , we have
$x=r\;\cos\alpha\;\;\;\;\;\;\;\;and\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;y=r\;sin\alpha$

equations (1) and (2) may be re-written as:

$X=r\;\cos\alpha\;\cos\theta+\;r\;\sin\alpha\;\sin\theta\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(3)$ .

$Y=r\;sin\alpha\;\cos\;\theta\;+\;r\;\cos\alpha\;\sin\;\theta\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(4)$

Substituting $x=r\;\cos\alpha\;\;\;and\;\;\;y=r\;sin\alpha$ , we get

$X=x\cos\theta+y\sin\theta\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(5)$ .

$Y=y\cos\theta-x\sin\theta\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(6)$ .

$i.e,\;\;\;\;\;\;(X,Y)\;=\;(x\cos\theta+y\sin\theta\;\;\;,\;\;\;y\cos\theta-x\sin\theta\;)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ .

are the coordinates of P referred to the new axes $OX$ and $OY$ .

Example Rotation of Axis:

The $xy$ -coordinate axes are rotated about the origin through an angle of $30^\circ$ If the $xy$ -coordinates of a point are $(5, 7)$ , find its $XY$ -coordinates, where $OX$ and $OY$ are the axes obtained after rotation.

Solution:

Let $(X, Y)$ be the coordinates of $P$ referred to the $XY$ -axes.
since, we have

$X=x\cos\theta+y\sin\theta$ .

$Y=y\cos\theta-x\sin\theta$ .

using values, $(x,y)=(5,7) \;\;\;\;\;\;and\;\;\;\;\;\; \theta = 30^\circ$ , we get

$X=5\cos30^\circ+7\sin30^\circ$ .

$Y=7\cos30^\circ-5\sin30^\circ$ .

So, $X={\textstyle\frac{5\sqrt3}2}+{\textstyle\frac72}\;\;\;\; and\;\;\;\;\; Y={\textstyle\frac{7\sqrt3}2}-{\textstyle\frac52}$ .

$i.e, \;\;\;\;\;\boxed{(X,Y)=({\textstyle\frac{5\sqrt3}2}+{\textstyle\frac72}\;\;,{\textstyle\frac{7\sqrt3}2}-{\textstyle\frac52})}$ .

Rotation of Axes Example:

The $xy$ -axes are rotated about the origin through an angle of $\arctan\frac43$ lying in the first quadrant. The coordinates of a point $P$ referred to the new axes $OX$ and $OY$ are $P (-1, -7)$ . Find the coordinates of $P$ referred to the $xy$ -coordinate system.
Solution: