Chapter 11: Parametric Equations and Polar Coordinates - Section 11.1 - Parametrizations of Plane Curves - Exercises 11.1 - Page 647: 19

a) $x=a \cos t; y= -a\sin t$; $0\leq t\leq 2 \pi$ b) $x=a \cos t; y= a\sin t$; $0\leq t\leq 2 \pi$ c) $x=a \cos t; y= -a\sin t$; $0\leq t\leq 4 \pi$ d) $x=a \cos t; y= a\sin t$; $0\leq t\leq 4 \pi$

The equation of a circle is $x^2+y^2=a^2$ at point $(a,0)$ with parametric equations such as: $x=a \cos t; y= a\sin t$ These parametric equations tells us the following some points: I. When we need to trace the circle clockwise, the parametric equation changes from $y= a\sin t$ to $y=- a\sin t$. This leads to the explanation given in the parts (a) and (c). II. When we need to trace the circle twice times, we will have to double the angle from $2\pi$ radians to $4 \pi$ radians so that $0\leq t\leq 4 \pi$. This leads to the explanation given in the parts (c) and (d). III. When we need to trace the circle counterclockwise to complete a one around of a circle, we will have $2\pi$ radians so that $0\leq t\leq 2 \pi$. This leads to the explanation given in the part (b).