Question

In Exercises \(5-22,\) a parametrization is given for a curve. (a) Graph the curve. What are the initial and terminal points, if any? Indicate the direction in which the curve is traced. (b) Find a Cartesian equation for a curve that contains the parametrized curve. What portion of the graph of the Cartesian equation is traced by the parametrized curve? $$x=4 \sin t, \quad y=5 \cos t, \quad 0 \leq t \leq 2 \pi$$

Short answer

The curve is an ellipse with an equation \((x/4)^2 + (y/5)^2 = 1\), traced in a counter-clockwise direction. The entire ellipse is traced since \(t\) ranges from \(0\) to \(2\pi\). The initial and terminal points are both at \((0, 5)\).

Step-by-step solution

Graph the Parametrized Curve and Determine Key Points

Our parametric equations are \(x = 4 \sin t\) and \(y = 5 \cos t\) for \(0 \leq t \leq 2\pi\). If we select values for \(t\) within this range and use these to calculate corresponding \(x\) and \(y\) values, we can sketch the graph of the curve. The curve starts at \(t = 0\) (the initial point) and ends at \(t = 2\pi\) (the terminal point). These points should be marked on the graph, as well as the direction in which the curve is traced.

Find the Cartesian Equation

A Cartesian equation can be found by eliminating the parameter \(t\). This can be achieved by using the trigonometric identity \(\sin^2 t + \cos^2 t = 1\). In our case, the identity will look like \((x/4)^2 + (y/5)^2 = 1\), which represents an ellipse.

Determine the Traced Portion of the Cartesian Graph

With the given range for \(t\) (from \(0\) to \(2\pi\)), the entire ellipse, as described by the Cartesian equation \((x/4)^2 + (y/5)^2 = 1\), is traced out by the parametrized curve. This can be seen from the graph created in Step 1.