Problem 45 - Precalculus Mathematics for Calculus

Question

Use a graphing device to draw the curve represented by the parametric equations. $$x=3 \sin 5 t, \quad y=5 \cos 3 t$$

Step-by-Step Solution

Answer

Graph \(x=3\sin 5t, \quad y=5\cos 3t\) using a parametric graph to see a looping pattern.

Step 1: Understand the Parametric Equations

You are given two parametric equations: \(x=3\sin 5t\) and \(y=5\cos 3t\). In these equations, \(t\) is a parameter that usually represents time. These equations describe a path traced by a point in the Cartesian plane as \(t\) varies.

Step 2: Determine the Range of t

For this exercise, we need to decide the interval for \(t\) to fully plot the curve. A common interval used is \(0\leq t \leq 2\pi\), covering a complete cycle of sine and cosine. However, check your graphing device or assignment requirements to confirm this.

Step 3: Set Up and Use a Graphing Device

Enter the equations into your graphing device. Each device will have a different method, but typically you will input them as a parametric curve, setting the equation for \(x\) and \(y\) with \(t\) as the parameter. Ensure you've configured the range for \(t\) correctly.

Step 4: Analyze the Graphed Curve

The graph will display a path that the point follows as \(t\) varies from the set interval. Look for patterns such as symmetry or loops to describe the shape of the curve. In this case, since \(x\) and \(y\) are in terms of \(\sin\) and \(\cos\), the curve may have an intricate, looping shape.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental building blocks in mathematics, specifically dealing with angles and the relationships between them. They include sine (\(\sin\)) and cosine (\(\cos\)) functions, which are periodic and oscillate between -1 and 1. These functions play a crucial role in parametric equations as they help describe periodic motions and waves. For example, the given parametric equations in the exercise, \(x=3\sin 5t\) and \(y=5\cos 3t\), involve sine and cosine, which suggest the motion traced by these can be expected to repeat in a regular pattern.

• The sine function, \(\sin\), is responsible for vertical oscillation in parametric forms.
• The cosine function, \(\cos\), tends to influence horizontal movement.

When considering a range of \(t\) from \(0\) to \(2\pi\), these functions complete their full cycle, representing one complete wave along the Cartesian plane. The fact that they multiply by constants 3 and 5 in the equations signifies amplitude changes, which stretch or compress the wave, impacting the size of the curve in respective axes. Understanding sine and cosine is vital as they not only define the shape of the graph but determine its symmetry and periodicity as well.

Graphing Parametric Curves

Graphing parametric curves involves plotting a function defined by two or more parametric equations. These equations express coordinates of points, thus allowing the representation of more complex curves than Cartesian equations do. In our case, the curves defined by \(x=3\sin 5t\) and \(y=5\cos 3t\) create a path as the parameter \(t\) evolves over a specified domain.
To plot such curves, it's common to use a graphing device which can handle parametric inputs:

• Input parametric equations correctly, ensuring \(t\) is well-defined.
• Set your graph range for \(t\) appropriately, typically \(0\leq t\leq2\pi\) to view complete cycles.
• Analyze the graph for specific characteristics—like symmetry, orientation, and number of loops.

The use of both sine and cosine in the equations suggests a looped, possibly elliptical path. Therefore, observing how the path weaves across the plane can provide insights into the nature of the relationships between the parametric functions. Visualizing this can illuminate the beauty in the mathematics, revealing intricate patterns such as symmetrical loops due to periodic nature of trigonometric functions.

Cartesian Plane

The Cartesian plane, also known as the coordinate plane, is a two-dimensional plane formed by the intersection of a vertical line (the y-axis) and a horizontal line (the x-axis). Each point on this plane is defined by an \(x\)-coordinate and a \(y\)-coordinate, representing the horizontal and vertical distances from the origin, respectively.
Parametric equations often trace paths through this plane by moving a point around as the parameter \(t\) varies. When you graph the given parametric equations \(x=3\sin 5t\) and \(y=5\cos 3t\), you essentially instruct a point to move through positions determined by the sine and cosine functions.

• The amplitude of these trigonometric functions affects how far the curve extends on the plane: 3 along the \(x\)-axis, 5 along the \(y\)-axis.
• The periodic nature of sine and cosine implies the figure traced will loop back upon itself.

Understanding how these coordinates work in the Cartesian plane is fundamental to graphing and interpreting curves. Each complete cycle of \(t\) consequently creates familiar shapes, often revealing symmetries and patterns that can only be elegantly expressed parametric equations.