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Expert-verified Found in: Page 860 ### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861 # 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.$r\left(t\right)=\left(1+\mathrm{sin}t,3-\mathrm{cos}2t\right),fort\in \left[0,2\mathrm{\pi }\right]$

The parametric equation of the vector valued functions $r\left(t\right)=\left(1+\mathrm{sin}t,3-\mathrm{cos}2t\right),fort\in \left[0,2\mathrm{\pi }\right]$ is $y=2{x}^{2}-4x+4$.

And the graph of the function is: See the step by step solution

## Step 1. Given Information.

The function:

$r\left(t\right)=\left(1+\mathrm{sin}t,3-\mathrm{cos}2t\right),fort\in \left[0,2\mathrm{\pi }\right]$

## Step 2. Find the parametric equations.

The parametric equations for the given function is:

$x=1+\mathrm{sin}t\phantom{\rule{0ex}{0ex}}y=3-\mathrm{cos}2t$

We know that,

$\mathrm{cos}2t=1-2{\mathrm{sin}}^{2}t$

From the given x, y,

$x-1=\mathrm{sin}t\phantom{\rule{0ex}{0ex}}\mathrm{cos}2t=3-y$

So,

$\mathrm{cos}2t=1-2{\mathrm{sin}}^{2}t\phantom{\rule{0ex}{0ex}}3-y=1-2{\left(x-1\right)}^{2}\phantom{\rule{0ex}{0ex}}y=2{x}^{2}-4x+4$

## Step 3. Find the ordered pairs.

The ordered pairs of the function is:

 t x=1+sin t y=3-cos2t (x, y) $0$ $1$ $2$ $\left(1,2\right)$ $\frac{\mathrm{\pi }}{2}$ $2$ $4$ $\left(2,4\right)$ $\mathrm{\pi }$ $1$ $2$ $\left(1,2\right)$ $\frac{3\mathrm{\pi }}{2}$ $0$ $4$ $\left(0,4\right)$ $2\mathrm{\pi }$ $1$ $2$ $\left(1,2\right)$

## Step 4. Graph the function.

The graph of the function is:  ### Want to see more solutions like these? 