# How do you find the parametric equations of a circle?

We'll start with the parametric equations for a circle:

#y = rsin t#

#x = rcos t#

where

If you know that the implicit equation for a circle in Cartesian coordinates is

We will take the equation for

#x/r = cos t#

#t = arccos (x/r)#

Now substitute into the equation for

#y = rsin arccos(x/r)#

Thus,

#y = r*sqrt(r^2 - x^2)/r#

This simplifies to

#y = sqrt(r^2 - x^2)#

If we square this entire deal and solve for

#r^2 = x^2 + y^2#

which is precisely the equation for a circle in Cartesian coordinates.

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To find the parametric equations of a circle, you can use the parametric form:

[x = x_c + r \cdot \cos(\theta)] [y = y_c + r \cdot \sin(\theta)]

where: (x_c) and (y_c) are the coordinates of the center of the circle, (r) is the radius of the circle, and (\theta) is the parameter, typically ranging from (0) to (2\pi) for a full circle.

By varying (\theta) from (0) to (2\pi), you can generate points along the circumference of the circle.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- What is the arc length of the curve given by #r(t)= (1,t,t^2)# on # t in [0, 1]#?
- What is the arclength of #f(t) = (sin2t-tcsct,t^2-1)# on #t in [pi/12,(5pi)/12]#?
- What is the arclength of #f(t) = (te^(2t)-e^t-3t,-2t^2)# on #t in [1,3]#?
- How do you find parametric equations for the tangent line to the curve with the given parametric equations at the specified point #x = 1+10 * sqrt(t)#, #y = t^5 - t#, and #z=t^5 + t# ; (11 , 0 , 2)?
- What is the arc length of the curve given by #r(t)= (ln(1/t),t^2,t)# on # t in [1, 10]#?

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