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Magazine Chapter 7: Problem 8
For the following exercises, use technology (CAS or calculator) to sketch the parametric equations. $$ [\mathbf{T}] x=3 \cos t, \quad y=4 \sin t $$
Short Answer
Expert verified
The parametric equations describe an ellipse centered at the origin with a horizontal radius of 3 and a vertical radius of 4.
Step by step solution
01
Identify the Parameterization
The given parametric equations are expressed in terms of the parameter \( t \). The equations provided are \( x = 3 \cos(t) \) and \( y = 4 \sin(t) \). These equations represent an ellipse, with \( t \) varying typically from \( 0 \) to \( 2\pi \).
02
Rewrite in Standard Form
The equations can be rewritten in a way that resembles the equation of an ellipse. Recall that the standard form of an ellipse is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here, \( a = 3 \) and \( b = 4 \). So, setting \( x = 3 \cos(t) \) and \( y = 4 \sin(t) \), squaring both sides gives \( \frac{x^2}{9} + \frac{y^2}{16} = 1 \).
03
Set Up Graphing Tool
To sketch the graph, use a graphing calculator or computer algebra system (CAS). Enter the parametric equations \( x = 3 \cos(t) \) and \( y = 4 \sin(t) \). Ensure that the tool is in parametric mode if necessary, and set the interval for \( t \) from \( 0 \) to \( 2\pi \).
04
Interpret the Graph
After plotting the equations using the tool, observe the shape formed. The graph should outline an ellipse centered at the origin, with a horizontal radius of 3 and a vertical radius of 4. This visualization confirms the mathematical conversion from parametric form to the standard form of an ellipse.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Geometry of an Ellipse
An ellipse is a fascinating shape in geometry, often described as an elongated circle. This shape has two focal points, and any point on the ellipse maintains a consistent sum of distances to these foci.
Unlike circles that have a single radius, ellipses have two axes, the major axis (longest diameter) and the minor axis (shortest diameter). In the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), the segments \( a \) and \( b \) represent half the lengths of the major and minor axes, respectively.
Unlike circles that have a single radius, ellipses have two axes, the major axis (longest diameter) and the minor axis (shortest diameter). In the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), the segments \( a \) and \( b \) represent half the lengths of the major and minor axes, respectively.
- The major axis is aligned along the coordinate with the larger denominator (either \( a^2 \) or \( b^2 \)).
- The center of the ellipse is the midpoint between the foci, positioned at the origin for equations in standard form.
Parameterization and its Role
Parameterization involves expressing a set of equations as functions of a parameter, which allows for a more flexible description of curves. In our case, the parameter is \( t \), which usually varies from \( 0 \) to \( 2\pi \).
- Using parametric equations like \( x = 3 \cos(t) \) and \( y = 4 \sin(t) \), we define the position of each point on the ellipse based on the angle \( t \).
- This method is particularly beneficial for representing curves that are not functions of \( x \) or \( y \), such as ellipses and circles.
Using a Graphing Calculator
Graphing calculators simplify the graph-drawing process for parametric equations by providing an intuitive interface. To sketch the ellipse given by \( x = 3 \cos(t) \) and \( y = 4 \sin(t) \), follow these steps:
- Ensure that the calculator is in parametric mode. This mode allows input of equations dependent on a parameter, here \( t \).
- Enter the parametric equations into the calculator. Set the parameter \( t \) to range between \( 0 \) and \( 2\pi \) for a complete sketch.
- Carefully adjust the window settings to accommodate the ellipse鈥檚 dimensions, ensuring both axes display the lengths correctly (horizontal axis showing up to 3, and vertical axis up to 4).
- Press the graph button to display the ellipse, confirming the symmetry and dimension align exactly with the expected major and minor axes.
Exploring a Computer Algebra System (CAS)
A Computer Algebra System offers powerful tools for solving, visualizing, and interpreting mathematical problems. With a CAS, you can input and analyze parametric equations like \( x = 3 \cos(t) \) and \( y = 4 \sin(t) \) with ease.
- Enter the parameterized equations directly into the CAS for immediate graphing.
- The system can automatically suggest the best view to see the whole ellipse or you can customize the view to see the desired level of detail.
- CAS can perform symbolic manipulation to convert the parametric form into the standard form, deepening understanding of the mathematical relationships.
- Beyond graphing, CAS can integrate, derive, and perform advanced operations on these functions, providing a more comprehensive toolkit for learners.
