Chapter 11, Problem 6RCC

To determine

To find: The relation between the coordinates (x, y) and (X, Y) when x-y axis are rotated through an angle.

Answer to Problem 6RCC

The desired relation between the coordinates is $y=X\sin \phi +Y\cos \phi$

Explanation of Solution

Given information :

x and y axis are rotated through an acute angle $\phi$ to produce the X and Y axis.

Graph

Consider x-y axis rotated through angle to obtain new axis $x^1-y^1$

Consider a point p(x, y)

Let (x, y) be old co-ordinates and (X,Y) be the new shifted co-ordinates ,

PA=y, PB=x, PC=y, PD=x

Now,

  $\begin{array}{l}PF=Y\cos \theta ,FC=AE=Y\sin \theta \\ OE=OC\cos \phi =OP\cos \phi =\cos \theta \end{array}$

Now, BP=x=OA. This can be written as:

OA=OE-AE

  $x=X\cos \phi -Ysin\phi$

Similarly, OB=y=PA. this can be written as: FA+FB

  $y=X\sin \phi +Y\cos \phi$

Interpretation :

The desired relation between the coordinates is $y=X\sin \phi +Y\cos \phi$