Question

In Problems 43 and 44, the parametric equations of four curves are given. Graph each of them, indicating the orientation.

$$\begin{gathered} C_1:x=t,y=\sqrt{1-t^2},-1\le t\le 1 \\ {C}_2:x=\sin t,y=\cos t,0\le t\le 2\mathrm{\pi } \\ {C}_3:x=\cos t,y=\sin t,0\le t\le 2\mathrm{\pi } \\ {C}_4:x=\sqrt{1-t^2},y=t,-1\le t\le 1 \end{gathered}$$

Short answer

The graph of each figure is shown in respective figures.

Step-by-step solution

Step 1. Given Information

The given parametric equations are

$$\begin{gathered} C_1:x=t,y=\sqrt{1-t^2},-1\le t\le 1 \\ {C}_2:x=\sin t,y=\cos t,0\le t\le 2\mathrm{\pi } \\ {C}_3:x=\cos t,y=\sin t,0\le t\le 2\mathrm{\pi } \\ {C}_4:x=\sqrt{1-t^2},y=t,-1\le t\le 1 \end{gathered}$$

Step 2. Graph of C1

We have $x=t,y=\sqrt{1-t^2},-1\le t\le 1$

The coordinates are$\left(-1,0\right),\left(0,1\right),\left(1,0\right)$

Plot these points on the plane and join them to obtain the graph

Step 3. Graph of C2

We have $x=\sin t,y=\cos t,0\le t\le 2\mathrm{\pi }$

The coordinates are $\left(0,1\right),\left(1,0\right),\left(0,-1\right),\left(-1,0\right),\left(0,1\right)$

Plot these points on the plane and join them to obtain the graph

Step 4. Graph of C3

We have $x=\cos t,y=\sin t,0\le t\le 2\mathrm{\pi }$

The coordinates are $\left(1,0\right),\left(0,1\right),\left(-1,0\right),\left(0,-1\right),\left(1,0\right)$

Plot these points on the plane and join them to obtain the graph

Step 4. Graph of  C4

We have $x=\sqrt{1-t^2},y=t,-1\le t\le 1$

Plot these points on the plane and join them to obtain the graph