# Finding Coordinates on a Circle: Solving for Points After Rotation

• macarino
In summary, To find the coordinates of a point on a circle after a given rotation, use the parametric equations x= r sin(\theta)+ a and y= r cos(\theta)+ b. However, if \theta is measured clockwise starting from the top of the circle, swap sine and cosine in the equations.
macarino
1. I have a circle equation
$$(x+a)^2+(y+b)^2=r^2$$
And suppose we start from the topmost of the circle
2.I have an angle Theta
and if Theta is larger than 0 we move counterclockwise or we will move in the opposite direction if it is positive.
, how can I find out the coordinates of a point on the circle after the move

Thanks a lot :)

macarino said:
1. I have a circle equation
$$(x+a)^2+(y+b)^2=r^2$$
And suppose we start from the topmost of the circle
2.I have an angle Theta
and if Theta is larger than 0 we move counterclockwise or we will move in the opposite direction if it is positive.

You mean "negative" instead of this second "positive" don't you?

, how can I find out the coordinates of a point on the circle after the move
Thanks a lot :)
For $\theta$ between 0 and $\pi/2$, at least, You can construct a right triangle by drawing a line from (0,0) to the point and then from the point perpendicular to the right angle. The y-coordinate of the point is the "near side", the x-coordinate is the negative of the "opposite side".

I am sorry I am not a native, not understanding what you mean

In the picture below, I'd like to know what is P's coordinates given rotation angle alpha and a point O(0,a)

The circle equation is
(x+a)^2+(y+b)^2=r^2

Thanks
Regards

#### Attachments

• untitled.bmp
54 KB · Views: 455
You are taking $\theta$ to be 0 at the "top", (a,b+ r) and measuring it clockwise? If you were using the "standard", $\theta= 0$ to the right, (a+r, 0), and measuring counter clockwise, then you could use the parametric equations, $x= r cos(\theta)+ a$, $y= r sin(\theta)+ b$. The fact that you starting 90 degrees off that means you need to swap sine and cosine: $x= r sin(\theta)+ a$ and $y= r cos(\theta)+ a$. Now, taking $\theta= 0$ you can see that $x= r(0)+ a= a$, $y= r(1)+ b= b+ r$ as you want. Further taking $\theta= \pi/2$, we have $x= r(1)+ a= a+ r$, $y= r(0)+ b= b$ as wanted.

$x= r sin(\theta)+ a$, $y= r cos(\theta)+ b$.

## 1. How do I find the coordinates of a point on a circle after it has been rotated?

To find the coordinates of a point on a circle after rotation, you will need to know the original coordinates of the point, the angle of rotation, and the center of the circle. First, draw a line from the center of the circle to the point. Then, use the angle of rotation to rotate this line counterclockwise. Finally, use the length of this new line and the original coordinates to find the new coordinates of the point using trigonometry.

## 2. What is the formula for finding coordinates on a circle after rotation?

The formula for finding coordinates on a circle after rotation is (x', y') = (x * cosθ - y * sinθ, x * sinθ + y * cosθ), where (x, y) are the original coordinates, θ is the angle of rotation, and (x', y') are the new coordinates.

## 3. Can I use the same formula for finding coordinates on a circle after multiple rotations?

Yes, you can use the same formula for finding coordinates on a circle after multiple rotations. Simply plug in the new angle of rotation and the previously calculated coordinates into the formula to find the new coordinates.

## 4. What if the circle is not centered at the origin?

If the circle is not centered at the origin, you will need to first translate the circle so that it is centered at the origin. This can be done by subtracting the coordinates of the center of the circle from the coordinates of the original point. Then, you can use the formula for finding coordinates on a circle after rotation.

## 5. Is there a quicker way to find coordinates on a circle after rotation?

Yes, there is a quicker way to find coordinates on a circle after rotation. You can use the matrix multiplication method, where you create a transformation matrix using the angle of rotation and the center of the circle, and then multiply this matrix by the original coordinates to find the new coordinates. This method can be more efficient for multiple rotations or for finding coordinates on a circle with a center other than the origin.

• Precalculus Mathematics Homework Help
Replies
19
Views
1K
• Precalculus Mathematics Homework Help
Replies
6
Views
1K
• Precalculus Mathematics Homework Help
Replies
8
Views
2K
• Precalculus Mathematics Homework Help
Replies
5
Views
3K
• Precalculus Mathematics Homework Help
Replies
20
Views
2K
• Precalculus Mathematics Homework Help
Replies
24
Views
779
• Precalculus Mathematics Homework Help
Replies
16
Views
1K
• Precalculus Mathematics Homework Help
Replies
2
Views
2K
• Precalculus Mathematics Homework Help
Replies
5
Views
2K
• Precalculus Mathematics Homework Help
Replies
2
Views
2K