Question

Use technology (CAS or calculator) to sketch the parametric equations. $$ x=\sec t, \quad y=\cos t $$

Short answer

Use graphing software, input the parametric equations, and observe curves between vertical asymptotes.

Step-by-step solution

Understanding Parametric Equations

Parametric equations represent a set of related equations where each term is defined as a function of a separate variable called a parameter, in this case, \(t\). Here we have \(x = \sec t\) and \(y = \cos t\). This means \(x\) and \(y\) change according to the parameter \(t\).

Identify the Domain of the Parameter

For the parametric equations, let's determine the values of \(t\) for which the expressions are defined. Note that \(\sec(t) = \frac{1}{\cos(t)}\) is undefined when \(\cos(t) = 0\). Thus, \(t\) should not be an odd multiple of \(\frac{\pi}{2}\). The common domain for cosine and secant functions is \(t \in (0, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \frac{3\pi}{2}) \cup (\frac{3\pi}{2}, \frac{5\pi}{2}),\) etc.

Set up Technology for Graphing

Use a CAS (Computer Algebra System) or a graphing calculator. Input the parametric functions \(x(t) = \sec t\) and \(y(t) = \cos t\). Set the parameter \(t\) within the domain identified in Step 2. Typical tools like Desmos or GeoGebra can be useful here.

Sketch the Graph

Observe the output on the graphing tool. You will see a series of vertical lines (asymptotes) intersected by curves due to the periodic nature of secant and cosine, noting the undefined points at odd multiples of \(\frac{\pi}{2}\). Analyze the behavior of the function based on how secant diverges.

Key concepts

Secant Function

The secant function, denoted as \( \sec t \), is related to the cosine function. It is defined as the reciprocal of the cosine: \( \sec t = \frac{1}{\cos t} \). This means wherever the cosine function has a value, the secant function also exists, but when the cosine function hits zero, the secant becomes undefined. This leads to vertical asymptotes on the graph.

• Vertical asymptotes occur at odd multiples of \( \frac{\pi}{2} \).
• The secant function is periodic, with a period of \( 2\pi \).
• It has the same periodicity but behaves differently regarding its amplitudes, extending beyond the typical interval of the cosine function.

When graphing, the secant function tends to create upward and downward curves that open towards these asymptotes, giving it a distinctive wave-like pattern.

Cosine Function

The cosine function, represented as \( \cos t \), is one of the fundamental trigonometric functions, notable for its smooth, continuous wave that oscillates between -1 and 1. It's a key player in describing cycles and oscillations, such as those of light waves or sound.

• The graph of the cosine function is a continuous wave that repeats every \( 2\pi \).
• Unlike secant, it does not have asymptotes and is defined for all real \( t \).
• Positions at which the cosine equals zero are particularly significant in trigonometry.

The cosine curve is symmetrical, peaking at \( 1 \) when \( t = 0, \pm 2\pi, \pm 4\pi, \ldots \) and reaching its lowest point at \(-1\) at \( t = \pm \pi, \pm 3\pi, \ldots \). Understanding the cosine function is crucial for interpreting parametrics where it acts as a variable modulator.

Graphing Technology

Graphing technology refers to tools and software designed to visualize mathematical functions and relationships. These include calculators like the TI-series and software platforms such as Desmos or GeoGebra. They provide a visual aid in understanding complex equations by translating abstract concepts into visual graphs.

• To use graphing technology effectively, input the equations correctly (such as parametric equations).
• Set the parameter range based on their domains to avoid undefined points impacting the visual output.
• Observe plotted points, curves, and intercepts to understand the behavior of functions involved.

This technology is important for students, as it bridges the gap between theoretical math and observable phenomena, making abstract concepts like secant and cosine functions more accessible and engaging.

Parametric Curves

Parametric curves represent paths traced by a point in a coordinate plane as the parameter varies. Unlike regular functions, parametric equations express both \( x \) and \( y \) as functions of another variable, often making them suitable for complex curves and motions. In the exercise, we use \( x = \sec t \) and \( y = \cos t \).

• They allow for a more flexible representation of curves that cannot be described with a single function \( y = f(x) \).
• Email smoothing shapes, creating loops, multiple outputs for a single input, or curves intersecting themselves.
• Used in physics and engineering to model trajectories or paths where time \( t \) influences position.

To graph parametric curves properly, it's essential to understand the interdependencies between variables, manipulate graphing tools accurately, and set the correct parameter ranges to capture the entire shape and behavior of the curve.