Problem 12 - Contemporary Precalculus

Question

Find a viewing window that shows a complete graph of the curve.\(x=12 \cos t, \quad y=12 \sin 2 t, \quad 0 \leq t \leq 2 \pi$$x=12 \cos t, \quad y=12 \sin 2 t, \quad 0 \leq t \leq 2 \pi$$x=12 \cos t, \quad y=12 \sin 2 t, \quad 0 \leq t \leq 2 \pi\)

Step-by-Step Solution

Answer

Answer: The viewing window that shows the complete graph of the curve is x from -12 to 12 and y from -12 to 12.

Step 1: Determine the range of x and y

Start by finding the minimum and maximum values for x and y within the given range of t. The x-coordinate is defined by \(x=12 \cos t\). Since the range of cosine function is between -1 and 1, the minimum and maximum values of x can be found by multiplying the minimum and maximum values of the cosine function by 12. Hence, x ranges from -12 to 12. The y-coordinate is defined by \(y=12 \sin 2t\). The range of the sine function is also between -1 and 1, but since the argument inside the sine function is 2t, it will complete two full periods within the given range of t (from 0 to \(2\pi\)). Thus, the minimum and maximum values of y are still found by multiplying the minimum and maximum values of the sine function by 12. So, y ranges from -12 to 12.

Step 2: Determine the viewing window

With the found range of x and y values, we can now determine the viewing window that shows the complete graph of the curve. The viewing window is defined as a rectangular region in the xy-plane, where the minimum and maximum values of x form the horizontal bounds, and the minimum and maximum values of y form the vertical bounds. In this case, the viewing window is: - Horizontal bounds: x = -12 to x = 12 - Vertical bounds: y = -12 to y = 12 So, the viewing window that shows the complete graph of the curve is x from -12 to 12 and y from -12 to 12.

Key Concepts

Graphing

Graphing parametric equations involves plotting a set of points that form a curve in the coordinate plane. For the exercise at hand, the equations are given as:

• \( x = 12 \cos t \)
• \( y = 12 \sin 2t \)

To graph these equations, substitute different values of the parameter \( t \) into the equations to obtain corresponding \( x \) and \( y \) coordinates.
These coordinates are then plotted on the XY-plane to form the curve. The parameter \( t \) acts as a control to draw the path sequentially, starting from the lowest value of \( t \) provided, going to the highest.
This method helps illustrate the movement and shape of the curve as \( t \) varies, showcasing how the x and y values change interdependently.

Trigonometric Functions

Trigonometric functions such as sine and cosine are essential in parametric equations. They help model periodic and oscillatory phenomena, like waves, circles, and more complex curves.
In this problem:

• The x-component uses \( \cos t \), which repeats its cycle every \( 2\pi \).
• The y-component uses \( \sin 2t \), effectively doubling the frequency, completing its cycle at every \( \pi \).

Knowing these properties allows us to predict behavior across intervals in \( t \).
Due to these periodicities, the curve will exhibit symmetrical and predictable patterns, controlled by these trigonometric functions. This knowledge is critical when analyzing or predicting the trajectory of the graph of such parametric equations.
Understanding how to manipulate and apply trigonometric functions helps reveal complex relationships in parametric plots.

Viewing Window

The viewing window is crucial for correctly displaying the graph of parametric equations. It determines which part of the plane you see when graphing the equations. Setting it correctly ensures the entire curve is visible and centered in your graphing tool.
For the equations given, our calculated bounds are:

• Horizontal bounds: x from -12 to 12
• Vertical bounds: y from -12 to 12

These bounds are gleaned from the range of the trigonometric functions involved.
The cosine and sine functions, when multiplied by 12, limit the range from -12 to 12 for both x and y. Thus, our viewing window is symmetrical around the origin.
When setting up a graphing window, ensure that the chosen range accommodates all possible values of the parametric equations. Errors in window settings can lead to cropped or misleading plots that don't represent the entire curve.