Q. 26 - 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.

$r\left(t\right)=(\sin 3t, \cos 3t) for t\in [0,2\mathrm{\pi }]$

Step-by-Step Solution

Answer

The parametric equation of the vector valued functions $r\left(t\right)=(\sin 3t, \cos 3t) for t\in [0,2\mathrm{\pi }]$ is $x^2+y^2=1$.

And the graph of the function is:

Step 1. Given Information.

The function:

$r\left(t\right)=(\sin 3t, \cos 3t) for t\in [0,2\mathrm{\pi }]$

Step 2. Find the parametric equations.

The parametric equations for the given function is:

$$\begin{gathered} x=x\left(t\right) \\ =\sin 3t \\ y=y\left(t\right) \\ =\cos 3t \end{gathered}$$

Squaring on both sides we get,

$$\begin{gathered} x^2+y^2={\sin }^{2}3t+{\cos }^{2}3t \\ {x}^2+y^2=1 \end{gathered}$$

Step 3. Graph the function.

The equation represents the circle with center $(0,0)$.

So the graph of $r\left(t\right)$ is: