Q. 27 - Calculus

Question

Find parametric equations for each of the vector-valued functions in Exercises 26–34, and sketch the graphs of the functions, indicating the direction for increasing values of t.

$r\left(t\right)=(2-\sin t, 4+\cos t) for t\in [0,2\mathrm{\pi }]$

Step-by-Step Solution

Answer

The parametric equation of the vector valued functions $r\left(t\right)=(2-\sin t, 4+\cos t) for t\in [0,2\mathrm{\pi }]$ is ${(x-2)}^2+{(y-4)}^2=1$.

And the graph of the function is:

Step 1. Given Information.

The function:

$r\left(t\right)=(2-\sin t, 4+\cos t) for t\in [0,2\mathrm{\pi }]$

Step 2. Find the parametric equations.

The parametric equations for the given function is:

$$\begin{gathered} x=x\left(t\right) \\ x=2-\sin t \\ x-2=\sin t \\ y=y\left(t\right) \\ y=4+\cos t \\ y-4=\cos t \end{gathered}$$

Adding and squaring on both sides, 

$$\begin{gathered} {(x-2)}^2+{(y-4)}^2={\sin }^{2}t+{\cos }^{2}t \\ {(x-2)}^2+{(y-4)}^2=1 \end{gathered}$$

Step 3. Graph the function.

The equation represent the circle with radius $1$and the center $(2,4)$.

So the graph of $r\left(t\right)$ is: