Problem 43 - Precalculus Mathematics for Calculus

Question

Use a graphing device to draw the curve represented by the parametric equations. $$x=\sin t, \quad y=2 \cos 3 t$$

Step-by-Step Solution

Answer

Graph the equations from \(t = 0\) to \(2\pi\) to see the curve.

Step 1: Understand the Parametric Equations

The parametric equations given are \(x = \sin t\) and \(y = 2 \cos 3t\). Here, \(t\) is the parameter that varies, and as it varies, it traces out a path that defines the curve or graph.

Step 2: Identify the Range of Parameter

To visualize the graph, identify a reasonable range for \(t\). For trigonometric functions, a common range is from \(0\) to \(2\pi\), which represents one complete cycle of the sine and cosine functions.

Step 3: Graph the Equations

Using a graphing device, such as a graphing calculator or software, input the parametric equations. Set the parameter \(t\) to vary from \(0\) to \(2\pi\). Adjust any settings to ensure the resolution is clear enough to see the graph's shape.

Step 4: Observe the Graph

Once the parametric equations are graphed, observe the resulting curve. The equation \(x = \sin t\) determines the x-coordinates as a sine wave oscillates between -1 and 1, while \(y = 2 \cos 3t\) creates vertical oscillations scaled by 2 with a frequency three times faster, causing interesting patterns.

Key Concepts

Graphing Parametric Equations

Parametric equations involve one or more variables called parameters that are used to express the coordinates of points on a curve. An intuitive way to understand parametric equations is to envision them as a set of instructions that trace a path on a coordinate plane. In the context of our exercise, we use the parameter \( t \) to determine the coordinates \( (x, y) \) where \( x = \sin t \) and \( y = 2 \cos 3t \). As \( t \) changes, these equations generate points which, when connected, form a curve.

To graph parametric equations effectively, follow these steps:

• Identify the parameter and how it influences each coordinate.
• Choose an appropriate range for the parameter to visualize one full cycle of the graph.
• Use a graphing tool to compute and plot the points.

Understanding parametric equations allows us to handle more complex curves easily, including those that describe intricate patterns and motions.

Trigonometric Functions

Trigonometric functions such as sine and cosine are foundational in describing periodic phenomena. They are invaluable in graphing parametric equations. In our exercise, the expressions \( x = \sin t \) and \( y = 2 \cos 3t \) utilize these functions.

Sine and cosine functions oscillate between -1 and 1, but they can be scaled and their frequencies adjusted to represent a wide variety of waveforms. For example:

• \( \sin t \) traces a waveform that cycles between -1 and 1.
• \( 2 \cos 3t \) modifies the wave to oscillate between -2 and 2, increasing its frequency by a factor of three.

Understanding how to manipulate these parameters allows you to predict and interpret the distinctive curve patterns resulting from parametric equations.

Graphing Calculators

Graphing calculators or software are indispensable when working with parametric equations, especially when they involve trigonometric functions that create intricate curves.

These devices provide you with a visual representation of the mathematical behavior of parametric equations, helping you understand:

• The ranges and cycles of parametric variables.
• The intricate details and complex interactions within a graph.

To use a graphing calculator effectively:

• Input the parametric equations accurately.
• Adjust settings such as the parameter range and display window to capture the full curve.
• Analyze the visual output to gain insights into the behavior of the equations.

Mastering a graphing calculator enhances your ability to comprehend and solve parametric equation problems.

Curve Tracing

Curve tracing is a method used to understand the form and behavior of curves described by parametric equations. This method involves visually interpreting the path traced by the equations as the parameter varies.

When you trace the curve for \( x = \sin t \) and \( y = 2 \cos 3t \), you observe the simultaneous behavior of both the x and y components:

• The x-component forms a sine wave that oscillates horizontally.
• The y-component, a cosine wave with triple frequency, creates rapid vertical oscillations.

This approach allows you to study complex motion patterns and gain insight into the mathematical properties of curves without needing extensive computation. It equips you with a deeper comprehension of interactions within the parametric framework, unveiling the harmony between different components of an equation.