Problem 12 - Technical Mathematics with Calculus

Question

Graph each pair of parametric equations. $$\begin{aligned} &x=\sin \theta\\\ &y=1.32 \sin 2 \theta \end{aligned}$$

Step-by-Step Solution

Answer

Plot the parametric equations by creating a table of \(\theta\), \(x\), and \(y\) values, plotting these points on a graph, and then connecting them to form the curve.

Step 1: Understand the parametric equations

The given parametric equations define the relationship between x and y for different values of the parameter \(\theta\). Here, \(x\) is given by the sine of \(\theta\), and \(y\) is given by 1.32 times the sine of twice \(\theta\).

Step 2: Set up a table of values

Create a table of values for \(\theta\), \(x\), and \(y\). Calculate the corresponding \(x\) and \(y\) for several values of \(\theta\) in its domain, which is usually from \(0\) to \(2\pi\) for a full cycle.

Step 3: Plot the points obtained from the table

Using the calculated x and y values from the table, plot the points on a graph. Be sure to consider the scale of the graph since the factor of 1.32 in the y-equation will stretch the graph vertically.

Step 4: Draw the curve

Connect the plotted points smoothly to reveal the shape of the curve that the parametric equations define. The pattern might repeat itself after a full cycle of \(\theta\) from \(0\) to \(2\pi\).

Key Concepts

Trigonometric Parametric Equations

When dealing with trigonometric parametric equations, it is essential to recognize how these equations enable us to define a curve by specifying the x and y coordinates separately concerning a third parameter, typically denoted as \theta (theta). This offers greater flexibility in curve representation, particularly for periodic functions like sine and cosine.

Consider the parametric equations in our example: \
x=\sin(\theta)\
and \(y=1.32 \sin(2\theta)\). Here, each variable depends on the sine function, which is inherently periodic, meaning it repeats values in a consistent cycle. As \(\theta\) varies, the x and y coordinates trace out a curve on the plane, where \(\theta\) often represents time or angle in applied scenarios.

To visualize the behavior of these equations, one can imagine a point moving along the curve as \(\theta\) increases. The sine function results in a smooth, wave-like motion for the x-coordinate, while the y-coordinate exhibits a similar pattern but with a different frequency and amplitude due to the scalar multiplier and the argument being multiplied by two.

Sine Function

The sine function is a fundamental trigonometric function that describes a smooth, periodic oscillation. In a unit circle, the sine of an angle \(\theta\) corresponds to the y-coordinate of a point on the circle's circumference. The function has a domain of all real numbers and a range of \([-1, 1]\), thus producing a wave-like graph when plotted over time.

In parametric equations, the sine function's ability to model periodic behavior is particularly useful. It can represent oscillations in mechanical systems, sound waves, alternating currents, and many other natural phenomena. Its graph is a smooth curve that oscillates above and below the x-axis, completing a full cycle every \(2\pi\) radians. This behavior is reflected in the parametric equation for x in our given example, \(x = \sin(\theta)\), indicating that as \(\theta\) changes, x simply traces out a sine wave.

Graphing in Polar Coordinates

Graphing in polar coordinates is another way to represent curves in the plane. Unlike Cartesian coordinates which use x and y coordinates, polar coordinates define a point's location based on its distance from the origin (radius r) and the angle (\(\theta\)) from the positive x-axis.

This system is particularly advantageous for curves that are circular or radial in nature, as the angle directly corresponds to the curve's rotation around the origin. When graphing parametric equations like our example, it's a good practice to consider if a polar representation can simplify the graphing process, or provide additional insights into the structure of the curve.

To convert the given parametric equations into polar form, one would typically isolate r and \(\theta\). Unfortunately, such a direct conversion isn't straightforward for our example, yet recognizing the periodic nature of the sine function in both x and y suggests the resulting curve will have symmetries and periodicity, characteristics often well-suited for representation in polar coordinates.

Parametric Curve Plotting

Parametric curve plotting allows for the representation of more complex curves that might not be expressible as a single function y=f(x). By defining x and y through parametric equations in terms of a third variable \(\theta\), we can plot a wide range of curves with varying shapes.

The process of plotting involves calculating x and y for a range of \(\theta\) values, often from 0 to \(2\pi\) for trigonometric functions that undergo a full cycle. One effective approach, highlighted in our step-by-step solution, is to create a table of \(\theta\) values and compute corresponding x and y coordinates.

After plotting these points, connecting them smoothly relies on understanding the curve's behavior, which can include direction, curvature, and any patterns or symmetries. The unique shape of the curve resulting from our parametric equations can reveal interesting features like loops, cusps, or multiple waves, illustrating how parametric plotting extends beyond simple graphs to capture intricate movements and relationships.