Question

For the following exercises, sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. $$ x=\sin \theta, y=1-\csc \theta, 0 \leq \theta \leq 2 \pi $$

Short answer

The Cartesian equation is \( y = 1 - \frac{1}{x} \), defined on \( x \in (-1, 1), x \neq 0 \).

Step-by-step solution

Understand the parametric equations

The parametric equations given are:- \( x = \sin \theta \)- \( y = 1 - \csc \theta \)where \( \theta \) ranges from 0 to \( 2\pi \). Our goal is to convert these equations into a single Cartesian equation and understand the shape of the curve.

Express \( y \) in terms of trigonometric functions

Knowing that \( \csc \theta = \frac{1}{\sin \theta} \), we can rewrite the equation for \( y \) as:\[y = 1 - \frac{1}{\sin \theta}\]Substituting \( x = \sin \theta \), the equation becomes:\[y = 1 - \frac{1}{x}\]This is the Cartesian form of the curve.

Determine the domain of the Cartesian equation

Since \( x = \sin \theta \), the range of \( x \) is \([-1, 1]\). However, \( \csc \theta \) is undefined when \( \theta \) is a multiple of \( \pi \), meaning \( x eq 0 \). Therefore, the domain for \( x \) is \(( -1, 1 )\) excluding 0.

Sketch or visualize the curve

Using the equation \( y = 1 - \frac{1}{x} \) over the domain \(( -1, 1 )\) (excluding \( x = 0 \)):- As \( x \) approaches 0 from either side, \( \frac{1}{x} \) grows large, meaning \( y \to -\infty \).- At \( x = 1 \), \( y = 0 \) and at \( x = -1 \), \( y = 2 \).The curve will show vertical asymptotic behavior near \( x = 0 \) and pass through the points (1,0) and (-1,2).

Key concepts

Cartesian Equation

In the realm of mathematics, a Cartesian equation is a single equation that relates two or more variables using a common coordinate system. For parametric equations like the ones given:

• \( x = \sin \theta \)
• \( y = 1 - \csc \theta \)

The aim is to eliminate the parameter, \( \theta \), to derive a relationship directly between \( x \) and \( y \). This conversion allows us to graph the equation on a simple two-dimensional plane without considering \( \theta \).
Knowing that \( \csc \theta = \frac{1}{\sin \theta} \), the expression for \( y \) becomes \( y = 1 - \frac{1}{x} \), thus connecting the variables \( x \) and \( y \) without \( \theta \). This Cartesian form provides a more straightforward way to analyze the curve's shape and properties.

Trigonometric Functions

Trigonometric functions like \( \sin \theta \) and \( \csc \theta \) play a key role in translating parametric equations into Cartesian form.

• \( \sin \theta \) describes the x-coordinate in this context, creating a familiar sine wave that varies from -1 to 1.
• The cosecant function, \( \csc \theta = \frac{1}{\sin \theta} \), inversely affects the y-coordinate, introducing complexity into the behavior of the curve due to division by zero issues when \( x = 0 \).

These functions define periodic phenomena, repeating their values over specific intervals. Understanding their behavior is essential to predict how parametric curves will behave over their specified range. These key properties ensure that the variable \( x \), derived through \( \sin \theta \), takes values within the open interval \( (-1, 1) \) since \( \csc \theta \) is undefined for \( \theta \) multiple of \( \pi \).

Curve Sketching

Curve sketching involves creating a visual representation of the relationship between variables in an equation. For the equation \( y = 1 - \frac{1}{x} \), the sketch reveals the overall shape and behavior of the curve.

• The domain \( ( -1, 1 ) \) excludes \( x = 0 \), indicating the presence of a vertical asymptote at this point. This means the curve will have infinite tendencies as it approaches this x-value, either from the left or the right.
• Key points like \( (1, 0) \) and \( (-1, 2) \) help in plotting significant parts of the curve, providing reference for its path and structure.

As \( x \) approaches zero, \( \frac{1}{x} \) increases dramatically, causing the curve to descend sharply, plummeting towards negative infinity. This concept of vertical asymptotes is crucial in understanding why the curve behaves drastically near \( x = 0 \). Clear visualization through curve sketching helps in grasping the finer details of the equation and its geometric implications, making it an invaluable tool in mathematical analysis.