Example 2 from Algebra and Trigonometry, 10.7 Parametric Equations: Graphs

Sketching the Graph of Trigonometric Parametric Equations

Construct a table of values for the given parametric equations and sketch the graph:

\begin{array}{l} x = 2 \cos t \\ y = 4 \sin t \end{array}

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution

Construct a table like that in Table 2 using angle measure in radians as inputs for $t,$ and evaluating $x$ and $y .$ Using angles with known sine and cosine values for $t$ makes calculations easier.

$t$	$x = 2 \cos t$	$y = 4 \sin t$
0	$x = 2 \cos (0) = 2$	$y = 4 \sin (0) = 0$
$\frac{π}{6}$	$x = 2 \cos (\frac{π}{6}) = \sqrt{3}$	$y = 4 \sin (\frac{π}{6}) = 2$
$\frac{π}{3}$	$x = 2 \cos (\frac{π}{3}) = 1$	$y = 4 \sin (\frac{π}{3}) = 2 \sqrt{3}$
$\frac{π}{2}$	$x = 2 \cos (\frac{π}{2}) = 0$	$y = 4 \sin (\frac{π}{2}) = 4$
$\frac{2 π}{3}$	$x = 2 \cos (\frac{2 π}{3}) = - 1$	$y = 4 \sin (\frac{2 π}{3}) = 2 \sqrt{3}$
$\frac{5 π}{6}$	$x = 2 \cos (\frac{5 π}{6}) = - \sqrt{3}$	$y = 4 \sin (\frac{5 π}{6}) = 2$
$π$	$x = 2 \cos (π) = - 2$	$y = 4 \sin (π) = 0$
$\frac{7 π}{6}$	$x = 2 \cos (\frac{7 π}{6}) = - \sqrt{3}$	$y = 4 \sin (\frac{7 π}{6}) = - 2$
$\frac{4 π}{3}$	$x = 2 \cos (\frac{4 π}{3}) = - 1$	$y = 4 \sin (\frac{4 π}{3}) = - 2 \sqrt{3}$
$\frac{3 π}{2}$	$x = 2 \cos (\frac{3 π}{2}) = 0$	$y = 4 \sin (\frac{3 π}{2}) = - 4$
$\frac{5 π}{3}$	$x = 2 \cos (\frac{5 π}{3}) = 1$	$y = 4 \sin (\frac{5 π}{3}) = - 2 \sqrt{3}$
$\frac{11 π}{6}$	$x = 2 \cos (\frac{11 π}{6}) = \sqrt{3}$	$y = 4 \sin (\frac{11 π}{6}) = - 2$
$2 π$	$x = 2 \cos (2 π) = 2$	$y = 4 \sin (2 π) = 0$

Table 2

Figure 3 shows the graph.

Graph of the given equations - a vertical ellipse.

Figure 3

By the symmetry shown in the values of $x$ and $y,$ we see that the parametric equations represent an ellipse. The ellipse is mapped in a counterclockwise direction as shown by the arrows indicating increasing $t$ values.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Graph the parametric equations: $x = 5 \cos t,$ $y = 3 \sin t .$

How did it go?