Problem 24 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) sketch the curve represented by the parametric equations (indicate the orientation of the curve). Use a graphing utility to confirm your result. (b) Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation, if necessary. $$x=\cos \theta, y=4 \sin \theta$$

Step-by-Step Solution

Answer

The graph of the parametric equations is an ellipse with a horizontal radius of 1 and a vertical radius of 4. The corresponding rectangular equation is \(x^2 + \frac{y^2}{16} = 1\).

Step 1: Sketching the Curve

To start, recognize the nature of these parametric equations. The equations given are \(x = \cos \theta\) and \(y = 4 \sin \theta\). They signify an ellipse in the x-y plane, whose horizontal radius is 1 (from \(\cos \theta\)) and vertical radius is 4 (from \(4 \sin \theta\)). Now, sketch this ellipse on a paper or use a graphing utility to plot it. Remember to indicate the orientation, which is counterclockwise (since as \(\theta\) increases, we move in a counterclockwise direction).

Step 2: Converting to Rectangular Form

To convert the parametric equations to a rectangular equation, one way is to solve one of the equations for \(\theta\) and then substitute into the other. However, for these particular equations, a more convenient way is to use the Pythagorean trigonometric identity \(\sin^2\theta + \cos^2\theta = 1\). Square both equations to get \(x^2 = \cos^2\theta\) and \(y^2 = 16\sin^2\theta\). Adding these two equations yields \(x^2 + \frac{y^2}{16} = 1\), which is the rectangular form of the given parametric equations. This equation represents an ellipse with a horizontal radius of 1 and a vertical radius of 4, consistent with our sketch. We don't need to adjust the domain in this case as the full ellipse is represented for \( \theta \) in the range from 0 to \(2\pi\).

Key Concepts

Elliptical Curves

Elliptical curves are fascinating geometrical shapes characterized by their oval-like appearance. Unlike circles, ellipses have two different radii: one for the horizontal axis and one for the vertical. In the given parametric equations

• \(x = \cos \theta\)
• \(y = 4 \sin \theta\)

these symbolize an ellipse. Here, the horizontal radius is 1 because the range of \(\cos \theta\) is from -1 to 1. The vertical radius is stretched to 4 due to the \(4 \sin \theta\). When sketching ellipses, observe these dimensions to get an accurate graph. This transformation affects the shape, making it elongated along one axis, depending on those scaling factors. Recognizing this relationship helps you understand the nature and formation of such geometrical curves.

Pythagorean Identity

In trigonometry, the Pythagorean identity is fundamental and can be a powerful tool when dealing with parametric equations. The identity \(\sin^2\theta + \cos^2\theta = 1\) originates from the Pythagorean theorem, which applies to right-angled triangles. This identity allows one to relate sine and cosine, which helps eliminate the parameter in equations and make them easier to work with.
By leveraging this identity, you can transition from parametric to rectangular forms smoothly. For example, by squaring the parametric equations \(x = \cos \theta\) and \(y = 4 \sin \theta\), you get \(x^2 = \cos^2 \theta\) and \(y^2 = 16 \sin^2 \theta\). Adding these gives us the relationship \(x^2 + \frac{y^2}{16} = 1\), revealing the rectangular form of the ellipse. Such transitions enhance understanding and provide a clear path to graphing complex entities.

Converting Parametric to Rectangular Equations

Converting parametric equations into rectangular form is a significant step in simplifying complex relations. The idea is to eliminate the parameter \(\theta\) and derive an equation in terms of \(x\) and \(y\) only. For the given parametric equations:

• \(x = \cos \theta\)
• \(y = 4 \sin \theta\)

Squaring gives \(x^2 = \cos^2 \theta\) and \(y^2 = 16 \sin^2 \theta\). By adding them and using the identity \(\cos^2 \theta + \sin^2 \theta = 1\), the rectangular equation \(x^2 + \frac{y^2}{16} = 1\) emerges. This new form captures the same ellipse but removes the placeholder \(\theta\), thus clarifying the relationship in a simpler expression.
This method often uncovers hidden features of the curve and is more applicable in various mathematical analyses.

Graphing Orientation

Understanding the orientation of a graph helps in correctly interpreting the parametric equations. The orientation indicates the direction in which a curve is traced as the parameter increases. For the parametric equations

• \(x = \cos \theta\)
• \(y = 4 \sin \theta\)

the orientation is counterclockwise. This is because as \(\theta\) goes from 0 to \(2\pi\), the curve describes a trajectory that goes in the opposite direction of clockwise.
To visualize this, imagine tracing the shape on the paper; as \(\theta\) increases, follow the path your pen takes. This can also be confirmed using graphing utilities that animate the tracing of curves, showing the path followed as you move along the curve. Being aware of graph orientation is crucial for accurate representation and understanding of parametric curves.