Find the parametric equations of a circle centered at the origin with radius of 4 where you start at point (-4,0) at s=0 and you travel counterclockwise with a period of 2π.
x = a cos s
y = a sin s
when s = 0, x = -4
so =-4 = a cos 0 = a
so
x = - 4 cos s
when s = pi/2 , y = -4
so -4 = a sin pi/2 = a
so y = -4 sin s
you know that
y = 4 sint
x = 4 cost
starts at (4,0)
You are 180° out of phase, starting at (-4,0)
So,
y = 4sin(t+π)
x = 4cos(t+π)
and Damon's equations work.
To find the parametric equations of a circle with radius 4 centered at the origin, we can use the trigonometric functions sine and cosine.
The general form of a parametric equation for a circle centered at the origin is:
x = r * cos(t)
y = r * sin(t)
where r is the radius of the circle and t is the parameter that varies.
For this specific case, where the radius is 4, the parametric equations become:
x = 4 * cos(t)
y = 4 * sin(t)
To start at point (-4, 0) when s = 0 and travel counterclockwise with a period of 2π, we need to adjust the parameter t accordingly.
The parameter t can be related to the arc length s by the equation:
s = r * t
Since s = 0 when t = 0, we need t to start at 0. However, we also want to start at the point (-4, 0), which corresponds to t = π. This means that we need to shift the values of t by π to account for the starting point.
To account for the period of 2π, we can multiply t by 2.
Therefore, the adjusted parametric equations for the circle become:
x = 4 * cos(2t + π)
y = 4 * sin(2t + π)