Find the parametric equations of a unit circle where you start at point (0,1) at t=0 and you travel clockwise with a period of 2π?
x(t)=?
y(t)=?
I know setting x(t)=cos(t) and y(t)=-sin(t) will make the circle travel clockwise but I'm not sure exactly on how I can make it start at the specific point (0,1)?
Any ideas anyone?
x=cos(t)
y=sin(t)
This will start at (1,0), but you want to start at (0,1)
Since cos(0) = 1, you need
y = cos(t)
x = sin(t)
Now you can see that as t->pi/2,
y goes down and x goes to the right. That makes it clockwise.
To find the parametric equations of a unit circle, we need to express the x and y coordinates of points on the circle as functions of a parameter, usually denoted as t.
In this case, we will represent the unit circle in a clockwise direction, starting at the point (0, 1) when t = 0. Since the period is 2π, after completing one full revolution, t will have a value of 2π again.
To describe the unit circle, we use the trigonometric functions sine and cosine. The x-coordinate of a point on the unit circle can be expressed as cosine(t), and the y-coordinate can be expressed as sine(t).
Therefore, the parametric equations for the unit circle starting at (0, 1) and traveling clockwise with a period of 2π are:
x(t) = cos(t)
y(t) = sin(t)
These equations represent the x and y coordinates of any point on the unit circle as a function of the parameter t.