Question

a circle of radius 5 centered at the point 2,1 and traced out counterclockwise

Answer 1

ok, what do you need to know about it?

Answer 2

find parameters

Answer 3

x = 2 + 5cos(t)
y = 1 + 5sin(t)
where t is between 0 and 2π

Answer 4

how did you come up with that can u please show me?

Answer 5

Circles are defined by the equation (x-a)^2 + (y-b)^2 = r^2 where (a, b) is the center and r is the radius.

Answer 6

i understand that you plugged in the points into the equation and you got (x-2)^2 + (y-1)^2 = 5^2 . i don't understand what happend next

Answer 7

Right. So if you remember that sin^2(x) + cos^2(x) = r^2 (a trig identity)

Answer 8

If we want to rewrite this in terms of parametric we want (x-2)^2 = sin^2(t) and (y-1)^2 = cos^2(t)
Solving each of these yields what I posted before and the 5 is just to factor for the radius.

Answer 9

can you please show me how u solved for one of them (cos)

Answer 10

The 5 is because if you want sin^2(t) + cos^2(t) [a trig identity that equals 1] to equal 25 (5^2) then you have to multiply the original equation by 25 so 25sin^2(t) + 25cos^2(t) = 5^2

(y-1)^2 = 25cos^2(t) so if you take the square root of both sides you'd get (y-1) = 5cos(t) and then adding 1 to both sides yields y = 1 + 5cos(t).

Answer 11

if it is counterclockwise should cos be equal to x

Answer 12

I think setting 0≤t≤2π handles the counterclockwise part (if you think about your unit circle this should make sense). I do know that parametric equations are not unique in that there are multiple ways to express the same shape with different parametric curves.

Answer 13

okay thank you for your help