Finding the (x,y) coordinates of arbitrary circles

In summary, the trig functions cos and sin can be used to find the (x,y) coordinates of a unit circle, but a general method for finding the coordinates of an arbitrary circle involves transforming the functions to the form of a * sin(bx+c) + d and a * cos(bx+c) + d. The values of a, b, c, and d can then be modified to account for the radius and center of the circle. This is essentially using polar coordinates in disguise.
  • #1
pakmingki
93
1
I have just started curiously thinking about this. The trig functions cos and sin give the (x,y) coordinates of the unit circle. How would i go about using the trig functions to finding (x,y) coordinates of an arbitrary circle?

What I am saying is, the cos and sin only work for the circle x^2 + y^2 = 1

I would like a general method for finding the (x,y) coordinates of the circle
(x-h)^2 + (y-k)^2 = r^2
I knwo this would involve simply transforming the original function of of a f(x)=sin(x) or f(x)=cos(x) to some form of f(x)= a * sin(bx+c) + d and f(x)= a * cos(bx+c) + d
However, i can't reason out what the values of a,b,c,d would be to transform a trig function to give the (x,y) coordinates of any arbitrary circle.
thanks.
 
Mathematics news on Phys.org
  • #2
Radius is a simple fix, consider the pythagorean identity

sin2t + cos2t = 1
Now let
x = cos(t) and
y = sin(t)

What do you get? Can you see how to modify this and get a circle of arbitrary radius?

Actually an arbitrary center is fairly easy as well, start with the equation of the circle and from there you should be able to see a similar substitution.
 
  • #3
well, actually i had bigger trouble with the circles with an arbitrary center.
For a radius, just simply modify the amplitude to the value of the radius.

but i still can't think of how to modify the functions for an arbitrary center
 
  • #4
In terms of x and y, how does a circle with arbitrary center at say (h,k) differ from a circle with a center at the origin? What do both equations look like? What changes?
 
  • #5
pakmingki said:
I have just started curiously thinking about this. The trig functions cos and sin give the (x,y) coordinates of the unit circle. How would i go about using the trig functions to finding (x,y) coordinates of an arbitrary circle?

firstly, there is something missing in the question... you said "trig fns cos and sin give the (x,y) coordinates of the unit circle"...?? How do they do that? They only give you the coordinates of a certain point on the unit circle if you have specified the angle! Now, once you have an angle and a radius (which is 1 in this case), you have effectively written everything in terms polar coordinates. Sin and cos are functions that helps you to tranform from one to the other...

ok, regarding your arbitrary circle business, it appears that you want a rather long-handed way to specify (x,y)... anyway... still doable I guess... assuming that you know the equation of the circle (then again why would u want to go all the way to express things in sin and cos?) in standard form
(x-h)^2 + (y-k)^2 = r^2, then simply ignore h and k for the moment and express your coordinates in terms of cos and sin of an angle then shift your answer by h and k respectively for the x and y coords.

Again this is just polar coordinates in disguise, this time you have also translated your origin relative to what you begin with.
 
  • #6
If the center of the circle is (a, b), just shift the values. If [itex](x-a)^2+ (y-b)^2= R^2[/itex] is the circle, for angle [itex]\theta[/itex], measured counter clockwise from the point (a+R,b), [itex]x= a+ Rcos\theta [/itex], [itex]y= b+ Rsin(\theta )[/itex].
 

1. How do I find the center point of a circle?

To find the center point of a circle, you will need to know the coordinates of any two points on the circle. The center point can be found by finding the midpoint between these two points. This can be done by adding the x-coordinates and dividing by 2 to find the x-coordinate of the center point, and adding the y-coordinates and dividing by 2 to find the y-coordinate of the center point.

2. How do I find the radius of a circle?

The radius of a circle can be found by measuring the distance from the center point to any point on the circle. This can be done using the distance formula, which is the square root of the sum of the squared differences between the x- and y-coordinates of the two points.

3. Can I find the (x,y) coordinates of a circle if I only know the radius and center point?

Yes, you can find the (x,y) coordinates of a circle if you know the radius and center point. Simply use the equation of a circle, which is (x - h)^2 + (y - k)^2 = r^2, where (h,k) is the center point and r is the radius. Plug in the values for h, k, and r, and you will have two possible coordinate pairs for the circle.

4. Can I use a graphing calculator to find the (x,y) coordinates of a circle?

Yes, most graphing calculators have a function for finding the (x,y) coordinates of a circle. This function is typically found under the "Geometry" or "Conics" menu. You will still need to input the necessary information, such as the radius and center point, to use this function.

5. Is there a formula for finding the (x,y) coordinates of a circle given three points on the circle?

Yes, there is a formula for finding the (x,y) coordinates of a circle given three points on the circle. This formula is called the "circumcenter formula" and can be found by solving a system of equations formed by the distance formula between the three points. However, this formula can be complex and it may be easier to use alternative methods, such as finding the center point and radius and using the equation of a circle.

Similar threads

  • General Math
Replies
5
Views
269
Replies
2
Views
1K
  • General Math
Replies
4
Views
758
Replies
2
Views
1K
Replies
1
Views
1K
  • General Math
Replies
4
Views
1K
  • General Math
Replies
2
Views
1K
Replies
1
Views
677
Replies
8
Views
828
Replies
4
Views
1K
Back
Top