Problem 16 - College Trigonometry

Question

In Exercises \(1-20\), plot the set of parametric equations by hand. Be sure to indicate the orientation imparted on the curve by the parametrization. $$ \left\\{\begin{array}{l} x=\sec (t) \\ y=\tan (t) \end{array} \text { for } \frac{\pi}{2}

Step-by-Step Solution

Answer

Plot the curve with vertical asymptotes at \( x = \pm \infty \) and indicate leftwards orientation from \( t = \frac{\pi}{2} \) to \( t = \frac{3\pi}{2} \).

Step 1: Understand the Parametric Equations

The parametric equations given are \( x = \sec(t) \) and \( y = \tan(t) \). The range for \( t \) is from \( \frac{\pi}{2} \) to \( \frac{3\pi}{2} \). These equations describe a curve in the Cartesian plane.

Step 2: Analyze Behavior of Secant and Tangent

In the range given, \( \sec(t) = \frac{1}{\cos(t)} \) and \( \tan(t) = \frac{\sin(t)}{\cos(t)} \). Note that in this interval, \( \cos(t) \) goes from 0 to -1 and back to 0, so \( x = \sec(t) \) will have vertical asymptotes at the endpoints \( t = \frac{\pi}{2} \) and \( t = \frac{3\pi}{2} \). Meanwhile, \( y = \tan(t) \) will show increasing behavior from negative infinity to positive infinity over this interval.

Step 3: Plot Key Points and Asymptotes

Identify key points and asymptotes in the interval: at \( t = \frac{\pi}{2} \), \( x \) and \( y \) are not defined; as \( t \) approaches \( \pi \), \( \cos(t) = -1 \) so \( x = -1 \) and \( y = 0 \). Finally, as \( t \) approaches \( \frac{3\pi}{2} \), \( x \) and \( y \) again approach infinity, and \( x \) becomes undefined due to the asymptote.

Step 4: Draw the Curve

On a Cartesian plane, draw vertical dashed lines (asymptotes) at \( x = \infty \) and \( x = -\infty \). Start plotting from the lower range towards higher values, tracing the curve along these key points. The curve moves from right to left because \( x \) decreases as \( t \) goes from just above \( \frac{\pi}{2} \) to approach \( \frac{3\pi}{2} \).

Step 5: Indicating the Orientation

Indicate the curve's orientation by drawing arrows on the curve. The arrows should point from larger values of \( x \) (positive infinity) to \( x = -1 \) at \( t = \pi \), continuing to \( x \) reaching negative infinity. This shows the direction of increasing \( t \).

Key Concepts

Secant Function

The secant function, denoted as \( \sec(t) \), is defined as the reciprocal of the cosine function: \( \sec(t) = \frac{1}{\cos(t)} \). This relationship means whenever the cosine function approaches zero, the secant function grows towards infinity. Consequently, the secant function has vertical asymptotes where \( \cos(t) = 0 \). These asymptotes appear at critical points such as \( t = \frac{\pi}{2} \) and \( t = \frac{3\pi}{2} \) for the interval given in the problem.

Important properties of the secant function to remember:

• Defined for all angles \( t \), except where \( \cos(t) = 0 \).
• As \( \cos(t) \) approaches zero, \( \sec(t) \) tends towards infinity.
• The function has periodic vertical asymptotes, aligning with the zeros of \( \cos(t) \).

In our specific case, \( \sec(t) \) contributes to the vertical asymptotes in the plotted curve, providing structure to the graph in the Cartesian plane.

Tangent Function

The tangent function, expressed as \( \tan(t) \), is defined by the ratio \( \tan(t) = \frac{\sin(t)}{\cos(t)} \). This definition leads to certain behaviors, particularly undefined points and asymptotic tendencies when \( \cos(t) = 0 \). The tangent function will exhibit vertical asymptotes at the points where \( \cos(t) = 0 \), similar to the secant function. However, the tangent function is bound to display unique characteristics in its behavior.

Key features of the tangent function include:

• The function is undefined at points where \( \sin(t) \) and \( \cos(t) \) are orthogonal (\( \cos(t) = 0 \)).
• As \( \cos(t) \) approaches zero from either direction, \( \tan(t) \) can increase or decrease towards infinity.
• The periodic pattern mirrors that of the sine and cosine functions, creating a cycle of undefined points and peaks.

In the given interval from \( \frac{\pi}{2} \) to \( \frac{3\pi}{2} \), the tangent function will be impacting the orientation of the curve as it transitions through these asymptotic events.

Asymptotes

Asymptotes are lines that a graph approaches but never touches, serving an essential role in understanding the behavior of certain functions. In the context of the secant and tangent functions, vertical asymptotes occur in places where the denominator of a fraction in the function approaches zero. This causes the function value to increase or decrease without bounds. More specifically:

• Vertical asymptotes appear at \( x = \infty \) or \( x = -\infty \) when \( \cos(t) = 0 \) in both \( \sec(t) \) and \( \tan(t) \).
• Between these asymptotes, the function will change dramatically, forming essential landmarks for graph drawing.

In parametric plotting, asymptotes help define the structure and directionality of curves. For instance, in this exercise, the graphs of \( \sec(t) \) and \( \tan(t) \) when combined create a path defined by these inherent boundaries.

Curve Plotting

Curve plotting with parametric equations involves visualizing a pair of equations simultaneously to form a pattern in the Cartesian plane. With parametric equations like \( x = \sec(t) \) and \( y = \tan(t) \), plotting requires understanding how each component evolves over a parameter \( t \).

The procedural steps for plotting such curves involve:

• Calculate values for \( x \) and \( y \) over the specified range of \( t \).
• Identify key points where significant changes happen, such as the function going to infinity.
• Locate the asymptotes to understand where the graph will have interruptions.
• Trace the curve ensuring the orientation is respected, marking direction using arrows.

For this particular problem, it’s about intuitively tracing the curve from right to left, considering the asymptotic boundaries and the function's tendencies as \( t \) progresses. This exercise clarifies the behavior of these complex functions in a visual and interactive way.