Let the -coordinate system be rotated through an angle , such that the range of the angle is about the same origin . The new coordinate system is -coordinate system as shown in the given diagram. Since triangle is a right triangle with so,
Since are the coordinates of the point with respect to the new coordinate system, -system, so .
Putting these values in (i) and (ii), we get the following:
Triangle is also a right triangle with
Putting these values in (iii) and (iv), we have the following equations as
These equations are used to find the coordinates of a point with respect to the new rotated coordinate system, -system. Thus, the point with respect to -plane is .
Conversely, if the coordinates of a point with respect -system are given, then its coordinates with respect to the original system can be determined by the equations
Example: The -coordinate axes are rotated about the origin through the angle of measure . The new axes are and . Find the -coordinates of the point .
Solution: Here and . The coordinates of with respect to the new -coordinates system are