Problem 17 - 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=\tan (t) \\ y=2 \sec (t) \end{array} \text { for }-\frac{\pi}{2}

Step-by-Step Solution

Answer

The curve is a hyperbola that opens vertically, symmetric about the y-axis, with its branches approaching vertical asymptotes at \(x = \pm 1\).

Step 1: Understand the Parametric Equations

We have two parametric equations: \(x = \tan(t)\) and \(y = 2 \sec(t)\). The parameter \(t\) ranges from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). Our task is to plot the curve described by these equations and indicate the direction in which the curve is traced as \(t\) increases.

Step 2: Analyze the Behavior of Tangent and Secant

Recall the behavior of the tangent and secant functions. As \(t\) approaches \(-\frac{\pi}{2}\) or \(\frac{\pi}{2}\), both \(\tan(t)\) and \(\sec(t)\) approach infinity. At \(t = 0\), \(\tan(t) = 0\) and \(\sec(t) = 1\), so the curve passes through the point \((0, 2)\).

Step 3: Identify Symmetry and Asymptotes

Since \(\tan(t)\) and \(\sec(t)\) are odd and even functions respectively, the parametric equations are symmetric about the y-axis. The curve will have vertical asymptotes at \(x = \pm 1\), corresponding to the limits of \(\tan(t)\).

Step 4: Plot Key Points

Calculate key points for some values of \(t\). At \(t = -\frac{\pi}{4}, -\frac{\pi}{8}, 0, \frac{\pi}{8}, \frac{\pi}{4}\), calculate \(x\) and \(y\) coordinates: - \(t = -\frac{\pi}{4}\): \(x = -1, y = 2\sqrt{2}\) - \(t = -\frac{\pi}{8}\): \(x = -\frac{1}{\sqrt{2}}, y = \frac{4}{\sqrt{2}}\) - \(t = 0\): \(x = 0, y = 2\)- \(t = \frac{\pi}{8}\): \(x = \frac{1}{\sqrt{2}}, y = \frac{4}{\sqrt{2}}\)- \(t = \frac{\pi}{4}\): \(x = 1, y = 2\sqrt{2}\)

Step 5: Sketch the Curve with Orientation

Using these points and noting the symmetry, plot the curve. Begin at the left side as \(t\) approaches \(-\frac{\pi}{2}\), trace it through the origin and out towards positive \(x\), ensuring to indicate the positive direction from left to right. The curve should resemble a symmetric hyperbola.

Key Concepts

Tangent Function

The tangent function, denoted as \( \tan(t) \), is an essential trigonometric function that appears frequently in parametric equations. It is defined as the ratio of the sine function to the cosine function: \[ \tan(t) = \frac{\sin(t)}{\cos(t)} \]. One of the primary characteristics of the tangent function is its periodic nature, repeating every \( \pi \) radians. However, for the interval \(-\frac{\pi}{2} < t < \frac{\pi}{2}\), it doesn’t complete a full period, as this range covers only one portion where it spans from negative to positive infinity.
- At \( t = 0 \), \( \tan(t) = 0 \) because \( \sin(0) = 0 \). - As \( t \) approaches \( \frac{\pi}{2} \) or \(-\frac{\pi}{2} \), \( \tan(t) \) approaches infinity as \( \cos(t) \) approaches zero.
This characteristic of shooting towards infinity results in the vertical asymptotes we observe on the graph. Knowing how the tangent function behaves helps to predict how the related parametric curve will look, especially near these boundaries.

Secant Function

The secant function, denoted as \( \sec(t) \), is closely related to the cosine function. It is defined as the reciprocal of the cosine function: \[ \sec(t) = \frac{1}{\cos(t)} \]. Since the cosine function can take on values from -1 to 1, the secant function can extend from negative to positive infinity, exhibiting an important property of being undefined wherever \( \cos(t) = 0 \).
- At \( t = 0 \), \( \sec(t) = 1 \) because \( \cos(0) = 1 \). - As \( t \) approaches \( \frac{\pi}{2} \) and \(-\frac{\pi}{2} \), \( \cos(t) \) approaches zero, causing \( \sec(t) \) to rise sharply towards infinity.
In the parametric equation \( y = 2 \sec(t) \), multiplying by 2 stretches the graph vertically, impacting the way the values change over this interval. Understanding the secant function's characteristics allows us to anticipate the vertical behavior of our parametric curve, especially at points where the function becomes unbounded.

Symmetry About the Y-Axis

Symmetry about the y-axis is a feature found in some curves, indicating that the curve is a mirror image about the line \( x = 0 \). In mathematical terms, when a curve is symmetric about the y-axis, for each point \( (x, y) \) on the curve, there is a corresponding point \( (-x, y) \).
For the parametric equations \( x = \tan(t) \) and \( y = 2 \sec(t) \), symmetry exists because:

• The tangent function \( \tan(t) \) is an odd function, satisfying \( \tan(-t) = -\tan(t) \).
• The secant function \( \sec(t) \), being even, satisfies \( \sec(-t) = \sec(t) \).

Together, these properties ensure that \( x \) changes sign while \( y \) does not, leading to a graph that reflects across the y-axis. Observing this symmetry is key when plotting parametric curves, as it provides an efficient way to predict and plot the curve behavior without calculating every single point.

Vertical Asymptotes

Vertical asymptotes are critical features in a curve where the value of the function heads toward infinity. They occur where the function is undefined or shoots rapidly to large values without bound.
In the context of this parametric curve defined by \( x = \tan(t) \) and \( y = 2 \sec(t) \), vertical asymptotes occur at values preventing \( t \) from approaching \( \frac{\pi}{2} \) and \(-\frac{\pi}{2} \), where \( \tan(t) \) and \( \sec(t) \) become unbounded.
- For the tangent function \( \tan(t) \), it grows towards \( \pm \infty \) near \( t = \pm \frac{\pi}{2} \), hence causing a vertical asymptote at those \( x \) values. - For the secant function, vertical asymptotes appear where \( \sec(t) \) diverges, aligning with the tangent's behavior and reinforcing these points of infinite stretch.
These asymptotes are indicative of the graph behaving like branches of a hyperbola, giving us critical insight into how to sketch such parametric curves correctly.