Solving Trig Transformations: Rotation of a Point

[smileandbehappy]

Homework Statement

A point with coordinates x and y is rotated in an anticlockwise direction through an angle z, so that its distance from the origin of the coordinate system remains constant. Show that the new x and y coordinates become

xcosz - ysinz and xcosz + ysinz respectively.

Homework Equations

The Attempt at a Solution

Well I am looking for some advice on how to start this! The wording is confusing to me! To my mind surely if you rotate a point it will be exactly the same? Any help would be supremely appretiated.

 

[Integral]

By rotating a point, they mean it is rotated about the origin by a angle z. start by drawing a picture, show the x and y components at each location. Now study the pic until you can identify the relationships.

 

[Architeuthis Dux]

I would approach this problem by first breaking down the given information and identifying the key components. The problem states that there is a point with coordinates x and y, and it is being rotated in an anticlockwise direction through an angle z. The important thing to note here is that the distance from the origin remains constant. This means that the point is not moving closer or further away from the origin, but rather it is being rotated around it.

To solve this problem, we can use trigonometric functions to find the new x and y coordinates. Since the point is being rotated in an anticlockwise direction, we can use the cosine and sine functions to find the new coordinates. The cosine function is used to find the new x coordinate, while the sine function is used to find the new y coordinate.

To understand how these functions work, we can visualize the point being rotated on a unit circle. As the point rotates, its x and y coordinates change based on the angle of rotation. The cosine and sine functions can be used to find these new coordinates.

Now, let's look at the equations given in the problem. The first one, xcosz - ysinz, represents the new x coordinate. We can break this down to understand how it relates to the problem. The "cosz" term represents the cosine of the angle z, while the "x" represents the original x coordinate of the point. Similarly, the "ysinz" term represents the sine of the angle z, while the "y" represents the original y coordinate. This equation essentially tells us that the new x coordinate is equal to the original x coordinate multiplied by the cosine of the angle z, minus the original y coordinate multiplied by the sine of the angle z.

Similarly, the second equation, xcosz + ysinz, represents the new y coordinate. Again, we can break this down to understand its relation to the problem. The "cosz" term represents the cosine of the angle z, while the "x" represents the original x coordinate. The "ysinz" term represents the sine of the angle z, while the "y" represents the original y coordinate. This equation tells us that the new y coordinate is equal to the original x coordinate multiplied by the cosine of the angle z, plus the original y coordinate multiplied by the sine of the angle z.

By using these equations, we can find the new x and y coordinates of the point