Look at the unit circle and apply the hint. The x- and y-coordinates of each point are cosθ and sinθ, respectively. The radius of the unit circle is, of course, 1, and the center point is (0, 0).
General form of the parametric equations for a circle:
x = r·cosθ+h
and
y = r·sinθ+k
where r is the radius and (h, k) is the center. Therefore, the parametric equations for the unit circle are
x = cosθ
and
y = sinθ
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The parametric equations
x = 2cosθ
and
y = 2sinθ
define a circle of radius 2, centered at the origin.
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The parametric equations
x = 4cosθ
and
y = 2sinθ
define a horizontal ellipse centered at the origin, with transverse axis of length 8 and conjugate axis of length 4.
:::::
If a = b then
x = a·cosθ
and
y = b·sinθ
define a circle centered at the origin.
If a > b, then
x = a·cosθ
and
y = b·sinθ
define a horizontal ellipse centered at the origin.
If a < b, If a = b then
x = a·cosθ
and y = b·sinθ
define a vertical ellipse centered at the origin.
