Problem 44 - Precalculus

Question

Use a graphing utility to graph the curve represented by the parametric equations. $$\begin{aligned} &x=\sec \theta\\\ &y=\tan \theta \end{aligned}$$

Step-by-Step Solution

Answer

The graph will show the curve represented by the given parametric equations. Since the secant function is undefined for some values of \(\theta\) and the tangent function takes all values, the graph might have some distinctive and irregular shapes.

Step 1: Identify the Parametric Equations

The equations given are \(x=\sec \theta\) and \(y=\tan \theta\). These are the parametric equations that will be used to plot the curve.

Step 2: Understand the Relationship

In these equations, \(\theta\) is the parameter. \(x\) and \(y\) are expressed in terms of trigonometric functions of \(\theta\). The secant function is the reciprocal of the cosine, and it varies from -∞ to -1 and from 1 to ∞. The tangent function is the ratio of the sine to the cosine function, and it takes all real values. This implies the graph will have a unique shape.

Step 3: Plug in the Parametric Equations into the Graphing Utility

The next step is to enter the parametric equations into a graphing utility. Set \(x=\sec \theta\) and \(y=\tan \theta\) in the graphing software. Make sure to set the mode of calculator to parametric mode if it's not by default.

Step 4: Graph the Parametric Equations

After plugging in the equations, graph the parametric function. Observe the curve that the equations trace out for increasing values of \(\theta\).

Step 5: Analyze the Graph

Now that you have a graph, analyze it. Remember that the secant function has vertical asymptotes when the cosine function equals zero and the tangent function takes all values, which can give the graph a unique, perhaps irregular, shape.

Key Concepts

Graphing Utility

A graphing utility is a powerful tool used to visualize mathematical functions and equations. It can be a software application or a physical calculator.
Graphing utilities make it easy to handle complex equations by plotting their graphs. To plot parametric equations like the ones here, you should know how to switch your graphing utility to parametric mode. This mode allows you to input values in terms of a parameter, like \( \theta \) in our equations.
To use a graphing utility effectively:

• Enter the equations into the appropriate fields.
• Adjust any settings to ensure you are in the correct mode (parametric in this case).
• Choose the range for your parameter \( \theta \) to ensure the graph captures the features of the equation.
• Plot the graph and observe the shape and behavior.

By using these steps, the graphing utility allows you to clearly view relationships and interactions between mathematical components. This can be especially useful when dealing with trigonometric functions, which often have complex behavior.

Trigonometric Functions

Trigonometric functions are essential tools in mathematics, involving functions of angles. The most common trigonometric functions are sine, cosine, and tangent, corresponding to \( \sin\theta \), \( \cos\theta \), and \( \tan\theta \) respectively.
They arise in right triangles, where the angle \( \theta \) helps determine the ratios of the triangle's sides.
In trigonometry:

• The sine of an angle is the ratio of the opposite side to the hypotenuse.
• The cosine of an angle is the ratio of the adjacent side to the hypotenuse.
• The tangent of an angle is the ratio of the opposite side to the adjacent side.

Understanding these ratios is critical when working with parametric equations that use trigonometric functions. Each function has its unique properties and behaviors, such as periodicity and defined ranges, playing an important role in how curves are graphed and analyzed.

Secant Function

The secant function, represented as \( \sec\theta \), is the reciprocal of the cosine function. This means \( \sec\theta = \frac{1}{\cos\theta} \). The secant function has some unique properties that distinguish it from other trigonometric functions.
Because it is the reciprocal of the cosine, the secant function is undefined whenever \( \cos\theta = 0 \), affecting the graph with vertical asymptotes at such points.
Key things to note about the secant function:

• It has a period of \( 2\pi \), like the cosine function.
• It takes values from minus infinity to minus one and from one to infinity.
• Its graph is known for having periodic spikes, which are the asymptotes.

In our exercise, \( x = \sec \theta \), so the behavior of x as \( \theta \) changes will align with the secant's behavior with asymptotes dividing sections of the curve, a key factor when plotting it in a graphing utility.

Tangent Function

The tangent function, \( \tan\theta \), is the ratio of the sine to the cosine functions, \( \tan\theta = \frac{\sin\theta}{\cos\theta} \). This function has its distinct characteristics marking a significant role in trigonometry and calculus.
The tangent function is periodic and takes every real number as its value, unlike sine and cosine.
When considering the tangent function, remember:

• It has a period of \( \pi \), which is half that of sine and cosine.
• Its graph has vertical asymptotes where \( \cos\theta = 0 \) because the function is undefined there.
• It covers all real numbers, rising and falling through infinity as \( \theta \) approaches the asymptotes.

In the context of our parametric equation \( y = \tan\theta \), the variation in \( y \) will reflect this uninhibited rise and fall across values, contributing to the distinctive patterns seen on the graph of the parametric equations involved.