Chapter 11.5, Problem 1E

Suppose the x- and y-axes are rotated through an acute angle ϕ to produce the new X- and Y-axes. A point P in the plane can be described by its xy-coordinates (x, y ) or its XY-coordinates (X, Y). These coordinates are related by the following formulas.

$\begin{array}{cc}x=\_\_\_\_\_\_\_\_\_\_& X=\_\_\_\_\_\_\_\_\_\_\\ y=\_\_\_\_\_\_\_\_\_\_& Y=\_\_\_\_\_\_\_\_\_\_\end{array}$

To determine

To fill: The statement “Suppose x and y axes are rotated through an acute angle $\varphi$ to produce the new X and Y axes. A point $P$ in the plane can be described by its xy-coordinates $\left(x,y\right)$ or its XY-coordinates $\left(X,Y\right)$. These coordinates are related by the following formulas.

$x=$ $y=$ $X=$ $Y=$.”

Answer to Problem 1E

Suppose x and y axes are rotated through an acute angle $\varphi$ to produce the new X and Y axes. A point $P$ in the plane can be described by its xy-coordinates $\left(x,y\right)$ or its XY-coordinates $\left(X,Y\right)$. These coordinates are related by the following formulas.

$x=X\cos \varphi -Y\sin \varphi$

$y=X\sin \varphi +Y\cos \varphi$

$X=x\cos \varphi +y\sin \varphi$

$Y=-x\sin \varphi +y\cos \varphi$.

Explanation of Solution

Definition used:

Definition 1:

“Suppose x and y-axes in a coordinate plane are rotated through an acute angle $\varphi$ to produce the X and Y axes. Then the coordinates of a point $\left(x,y\right)$ in xy-plane in terms of X and Y are,

$x=X\cos \varphi -Y\sin \varphi$ and $y=X\sin \varphi +Y\cos \varphi$.

Also, the coordinates of a point $\left(X,Y\right)$ in XY-plane in terms of x and y are,

$X=x\cos \varphi +y\sin \varphi$ and $Y=-x\sin \varphi +y\cos \varphi$.”

Calculation:

If the x and y-axes of a coordinate plane are rotates through an acute angle $\varphi$ then the new position of the coordinate axes are denoted by X and Y.

Now by the definition stated above, the coordinates of a point $\left(x,y\right)$ in xy-plane in terms of X and Y are given by,

$x=X\cos \varphi -Y\sin \varphi$ and $y=X\sin \varphi +Y\cos \varphi$.

Similarly, a point $\left(X,Y\right)$ in XY-plane in terms of x and y are given by,

$X=x\cos \varphi +y\sin \varphi$ and $Y=-x\sin \varphi +y\cos \varphi$.