Q 6 - Calculus

Question

Use the results of Exercise $3$3 to analyze the direction of

motion for the parametric curves given by the equations in

Exercises $5-8$

$x=\sin t,y=\cos t,t\in \mathrm{ℝ}$

Step-by-Step Solution

Answer

 The required answer is the direction of motion is clockwise.

Step 1: Given information

The parametric equations :

$x=\sin t,y=\cos t,t\in \mathrm{ℝ}$

Step 2: Find the parametric equations that move in different directions.

The function $x=\sin t$

Differentiate in terms of $t$

$$\begin{gathered} x^1\left(t\right)=\frac{d}{dt}\left(\sin t\right) \\ {x}^1\left(t\right)=\cos t \\ {x}^1\left(t\right)>0 \end{gathered}$$

The function $y=\cos t$

Differentiate in terms of $t$

$$\begin{gathered} y^{\text{'}}\left(t\right)=\frac{d}{dt}\left(\cos t\right) \\ {y}^1\left(t\right)=-\sin t<0 \end{gathered}$$

That $x^1\left(t\right)>0 and y\left(t\right)<0$

Step 3: Find the direction of motion for the parametric curves given by the equations

When $t=0$

$$\begin{gathered} (x,y)=(\sin t,\cos t) \\ (x,y)=(\sin 0,\cos 0) \\ (x,y)=(0,0) \end{gathered}$$

The value of $x$ increases as $t$ increases while the value of $y$ decreases

Therefore the direction of motion is clockwise.