Problem 13 - 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. $$ \left\\{\begin{array}{l} x=2 \cos (t) \\ y=\sec (t) \end{array} \text { for } 0 \leq t<\frac{\pi}{2}\right. $$

Step-by-Step Solution

Answer

The graph is a right-to-left curve moving upwards, starting at (2, 1) and directed towards (0, ∞).

Step 1: Understanding the Parametric Equations

The given parametric equations are \(x = 2\cos(t)\) and \(y = \sec(t)\). These equations describe the curve in the plane where both \(x\) and \(y\) are functions of the parameter \(t\), with \(0 \leq t < \frac{\pi}{2}\).

Step 2: Analyze the Range of t

Consider the interval \(0 \leq t < \frac{\pi}{2}\). During this interval, the cosine function \(\cos(t)\) ranges from 1 to 0, and the secant function \(\sec(t) = \frac{1}{\cos(t)}\) ranges from 1 to infinity. This affects the range of \(x\) and \(y\).

Step 3: Understand the x and y Values

As \(t\) approaches 0, \(x = 2\cos(t)\) approaches 2 and \(y = \sec(t)\) approaches 1. As \(t\) approaches \(\frac{\pi}{2}\), \(x\) approaches 0 (the limit of 2 times approaching 0) and \(y\) tends to infinity since \(\sec(t) = \frac{1}{\cos(t)}\).

Step 4: Determine Orientation

To determine the orientation, observe how \(x\) and \(y\) change as \(t\) increases from 0 to \(\frac{\pi}{2}\). We start at the point \((2, 1)\) when \(t = 0\) and move towards \((0, \infty)\) as \(t\) approaches \(\frac{\pi}{2}\). The orientation is upwards along this path.

Step 5: Sketch the Curve

To sketch the curve:- Start at the point \((2, 1)\) for \(t = 0\).- Plot several points by choosing values of \(t\) between 0 and \(\frac{\pi}{2}\), calculating \(x = 2\cos(t)\) and \(y = \sec(t)\) for those values.- Note that as \(t\) increases, \(x\) decreases to 0 while \(y\) sharply increases towards \(\infty\).- The curve should show a segment of a rectangular hyperbola moving from right to left, with the direction of increasing \(y\) (upwards) as \(t\) increases.

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos(t) \), is a fundamental trigonometric function. It measures the horizontal distance from the origin to a point on the unit circle corresponding to an angle \( t \). Just like its sibling functions, the cosine function plays a crucial role in describing periodic behaviors, such as waves and circular motion.

Key properties of the cosine function include:

• Range: The cosine function's values lie between -1 and 1 for any real number \( t \).
• Periodicity: \( \cos(t) \) is periodic with a period of \( 2\pi \), meaning it repeats its values every \( 2\pi \) units.
• Symmetry: It is an even function, symmetric with respect to the y-axis, which means \( \cos(-t) = \cos(t) \).

In the context of our parametric equations, the expression \( x = 2\cos(t) \) reflects how the "stretch" of the cosine wave interacts with a scaling factor, 2, affecting its amplitude.

Secant Function

The secant function, represented as \( \sec(t) \), is the reciprocal of the cosine function. That is, \( \sec(t) = \frac{1}{\cos(t)} \). This ratio has a deep geometrical meaning, representing the length of the line segment from the origin to a tangent line touching the unit circle at the angle \( t \).

Important aspects of the secant function to consider are:

• Undefined Points: It is undefined wherever the cosine function equals zero. This happens at odd multiples of \( \frac{\pi}{2} \).
• Range: While \( \cos(t) \) ranges from -1 to 1, \( \sec(t) \) takes values \([1, \infty)\) or \((\text{-}\infty, -1]\).
• Periodicity: It has a period of \( 2\pi \), sharing this property with cosine and sine.

In our exercise, as \( t \) shifts between 0 and \( \frac{\pi}{2} \), \( y = \sec(t) \) ranges from 1 to infinity, illustrating the function's rapidly increasing nature as \( t \) approaches the asymptote at \( \frac{\pi}{2} \).

Curve Orientation

Curve orientation is an essential concept when dealing with parametric equations as it defines the direction in which a curve is traced as the parameter \( t \) changes. In practical terms, it's like determining if you are walking forwards or backwards along a path.

To ascertain the orientation of the curve described by our equations \( x = 2\cos(t) \) and \( y = \sec(t) \):

• Start at \( t = 0 \), where \( (x, y) = (2, 1) \).
• As \( t \) increases toward \( \frac{\pi}{2} \), \( x \) decreases from 2 to 0, while \( y \) surges towards infinity.
• The path starts right (at \( (2, 1) \)) and ascends steeply as it moves left (towards \( (0, \infty) \)).

This progression indicates upwards orientation, with the points traced upwards as \( t \) progresses, providing a clear sense of direction along the curve.

Rectangular Hyperbola

A rectangular hyperbola is a specific type of hyperbola with noteworthy symmetry and geometric properties. It is defined by the equation \( xy = c \), where \( c \) is a constant. For parametric equations, parts of these hyperbolas can appear in sketches like an arch or quadratic segment.

The details of our parametric curve resemble a segment of a hyperbola due to the inversely proportional relationship between \( x = 2\cos(t) \) and \( y = \sec(t) \) over the given interval:

• As \( x \) decreases, \( y \) increases, displaying a negative correlation.
• This shape and orientation can mirror one segment of a hyperbola's branches, here set specifically in the positive quadrant with \( x \geq 0 \) and \( y \geq 0 \).

Such characteristics help identify the curve segment in question as a rectangular hyperbola, illustrating how parametric equations can map out complex curves effortlessly.