Problem 20 - Thomas Calculus

Question

\(x=\sec t\) (enter as 1\(/ \cos (t) \)), \(y=\tan t\) enter as \(\sin (t) / \cos (t) ),\) over a. \(-1.5 \leq t \leq 1.5\) b. \(-0.5 \leq t \leq 0.5\) c. \(-0.1 \leq t \leq 0.1\)

Step-by-Step Solution

Answer

In sections a, b, and c, both \(\sec t\) and \(\tan t\) remain continuous without asymptotes due to the absence of points where \(\cos t = 0\).

Step 1: Understand the Parametric Equations

We have two parametric equations: \(x = \sec t\) and \(y = \tan t\). The secant function is the reciprocal of the cosine function, so \(x = \sec t = \frac{1}{\cos t}\). The tangent function is the ratio of the sine to the cosine, so \(y = \tan t = \frac{\sin t}{\cos t}\).

Step 2: Determine the Function's Nature

The parametric equations are derived from the trigonometric identities involving the unit circle. Notice that \(\sec t\) and \(\tan t\) are undefined for \(t\) values where \(\cos t = 0\), which occur at odd multiples of \(\frac{\pi}{2}\). These points are where vertical asymptotes will appear in the range specified.

Step 3: Range Analysis for Part (a)

Given \(-1.5 \leq t \leq 1.5\), the cosine function does not equal zero within this interval, so there are no asymptotes. Evaluate function behavior across this range to find the output on the Cartesian plane.

Step 4: Sketch and Behavior for Part (a)

Plot points or analyze the function behavior over the interval \(-1.5 \leq t \leq 1.5\). Since \(\sec t\) and \(\tan t\) are continuous here, the curve will be similarly smooth, resembling a segment of a hyperbola.

Step 5: Range Analysis for Part (b)

For \(-0.5 \leq t \leq 0.5\), this interval does not contain points where \(\cos t = 0\). Thus, \(\sec t\) and \(\tan t\) are continuous, and we can determine the output on the Cartesian plane similarly to Part (a).

Step 6: Range Analysis for Part (c)

For \(-0.1 \leq t \leq 0.1\), the cosine is strictly positive and never zero, meaning both \(\sec t\) and \(\tan t\) will produce continuous outputs without asymptotes. This results in another smooth and continuous curve segment within the interval.

Key Concepts

Secant Function

The secant function is an important trigonometric function, commonly expressed as \( \sec t \), which can be defined as the reciprocal of the cosine function: \( \sec t = \frac{1}{\cos t} \). This function helps us understand relationships in trigonometry, especially when dealing with parametric equations. One key feature of the secant function is that it becomes undefined whenever \( \cos t = 0 \).

This results from dividing by zero, which occurs at odd multiples of \( \frac{\pi}{2} \). Therefore, vertical asymptotes can form in graphs where secant is involved. For practical purposes, in parametric equations, knowing where \( \sec t \) is undefined helps in anticipating these asymptotic behaviors. Without these points, the function \( \sec t \) can smoothly transition as it depends on the cosine curve, varying from -∞ to ∞ as cosine approaches zero.

Tangent Function

The tangent function, denoted by \( \tan t \), is another fundamental trigonometric function. It is defined as the ratio of sine to cosine: \( \tan t = \frac{\sin t}{\cos t} \). This makes the tangent function undefined at the same places where the secant function is, specifically, when \( \cos t = 0 \).

Tangent has a periodic nature, with a period of \( \pi \), and vertical asymptotes occur where it is undefined. The tangent function is particularly useful in describing the slope of angles and finding angles in right triangles or in oscillatory phenomena. When used in parametric equations, like \( y=\tan t \), it reflects how the sine and cosine interplay in affecting the y-component of a curve.

Vertical Asymptotes

Understanding vertical asymptotes is crucial in dealing with functions like secant and tangent. A vertical asymptote is a line that a graph approaches but never really touches or intersects. This phenomenon happens because the denominator of a rational function becomes zero, resulting in an undefined value.

In the context of the secant \( \sec t = \frac{1}{\cos t} \) and tangent \( \tan t = \frac{\sin t}{\cos t} \) functions, vertical asymptotes occur whenever \( \cos t = 0 \). These points happen at odd multiples of \( \frac{\pi}{2} \) and represent the boundaries in the graph where the values of the function would tend towards infinity. Recognizing these patterns helps in sketching parametric curves appropriately and avoiding points where a function might become undefined.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables. They are essential in simplifying and solving trigonometric expressions and are widely used in parametric equations.

Some fundamental identities include:

• Pythagorean Identity: \( \sin^2 t + \cos^2 t = 1 \)
• Secant Identity: \( 1 + \tan^2 t = \sec^2 t \)
• Reciprocal Identity for Secant: \( \sec t = \frac{1}{\cos t} \)
• Reciprocal Identity for Tangent: \( \tan t = \frac{\sin t}{\cos t} \)

In problems where \( x = \sec t \) and \( y = \tan t \), these identities are invaluable in computational simplifications. They also underpin the geometric nature of curves in parametric plots, ensuring students can predict and comprehend the behavior of such curves on a Cartesian plane.