Chapter 8 - Complex Numbers, Polar Equations, and Parametric Equations - Section 8.6 Parametric Equations, Graphs, and Applications - 8.6 Exercises - Page 406: 56

$x = cos~t$ $y = sin~t$ These parametric equations trace out the unit circle in the counterclockwise direction. $x = cos~t$ $y = -sin~t$ These parametric equations trace out the unit circle in the clockwise direction.

$x = cos~t$ $y = sin~t$ When t = 0: $x = cos~0 = 1$ $y = sin~0 = 0$ As t increases from $t = 0$ to $t = \frac{\pi}{2}$: $x$ decreases from $x = 1$ to $x=0$ $y$ increases from $y = 0$ to $y=1$ Thus, starting from the point $(1,0)$, the points $(x,y)$ trace out the unit circle in the first quadrant as t increases from $t=0$ to $t = \frac{\pi}{2}$. This means that the unit circle is being traced in the counterclockwise direction. $x = cos~t$ $y = -sin~t$ When t = 0: $x = cos~0 = 1$ $y = -sin~0 = 0$ As t increases from $t = 0$ to $t = \frac{\pi}{2}$: $x$ decreases from $x = 1$ to $x=0$ $y$ decreases from $y = 0$ to $y=-1$ Thus, starting from the point $(1,0)$, the points $(x,y)$ trace out the unit circle in the fourth quadrant as t increases from $t=0$ to $t = \frac{\pi}{2}$. This means that the unit circle is being traced in the clockwise direction.