Problem 14 - Precalculus: Functions and Graphs

Question

Find an equation in \(x\) and \(y\) whose graph contains the points on the curve \(C\). Sketch the graph of \(C\), and indicate the orientation. $$x=\csc t, \quad y=\cot t, \quad-\pi

Step-by-Step Solution

Answer

The equation is \( y = -\sqrt{x^2 - 1} \), with the graph tracing from left to right in the fourth quadrant.

Step 1: Understand Trigonometric Relationships

Recall that the cosecant and cotangent functions are related to sine and cosine. Specifically, \( \csc t = \frac{1}{\sin t} \) and \( \cot t = \frac{\cos t}{\sin t} \).

Step 2: Express Sine in Terms of x

Given \( x = \csc t \), write this as \( x = \frac{1}{\sin t} \). Thus, \( \sin t = \frac{1}{x} \).

Step 3: Express y in Terms of x

Since \( y = \cot t = \frac{\cos t}{\sin t} \) and \( \sin t = \frac{1}{x} \), we can substitute to get \( y = \cos t \cdot x \).

Step 4: Use Pythagorean Identity to Find cos(t)

Using the identity \( \sin^2 t + \cos^2 t = 1 \), substitute \( \sin t = \frac{1}{x} \) to find \( \cos t = \sqrt{1 - \sin^2 t} = \sqrt{1 - \frac{1}{x^2}} \).

Step 5: Substitute cos(t) in Expression for y

Substitute \( \cos t = \sqrt{1 - \frac{1}{x^2}} \) into \( y = \cos t \cdot x \) to get \( y = x \cdot \sqrt{1 - \frac{1}{x^2}} \), simplifying to \( y = \sqrt{x^2 - 1} \), taking into account the negative range of \( t \).

Step 6: Analyze Range of t for Orientation and Sketch

Given \(-\pi < t < 0\), \( x \) (which is \( \csc t \)) ranges from \(-\infty\) to \(-1^-\), and \( y \) is non-positive. The curve is traced from left to right as \( t \) increases, sketching an arc in the fourth quadrant.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in understanding periodic phenomena, often used to model waves and repetitive patterns. These functions include sine (\( \sin t \)), cosine (\( \cos t \)), and their reciprocals such as cosecant (\( \csc t = \frac{1}{\sin t} \)) and cotangent (\( \cot t = \frac{\cos t}{\sin t} \)). When working with parametric equations, these functions help to describe the movement of a point along a curve over time.

• \( \csc t \) represents the reciprocal of the sine function, helpful in expressing angles and their relative positions.
• \( \cot t \) is the ratio of cosine to sine, giving us another perspective in trigonometric contexts.

Understanding these relationships allows for transforming and manipulating parametric forms into Cartesian coordinates, aiding in graph representation.

Pythagorean Identity

The Pythagorean Identity is a key tool in trigonometry, expressing the fundamental relationship between sine and cosine:\[ \sin^2 t + \cos^2 t = 1 \]. It reminds us that for any angle, the sum of the squares of sine and cosine is always equal to one.
This identity helps solve equations where you can express one trigonometric function in terms of another. For instance, if you know \( \sin t = \frac{1}{x} \), substituting into the identity allows you to find \( \cos t = \sqrt{1 - \frac{1}{x^2}} \). This expression is critical when transitioning from parametric to Cartesian equations by eliminating the parameter \( t \).
Such identities streamline our calculations and make solving parametric equations more manageable.

Graphing Techniques

Graphing parametric equations requires understanding how components \( x(t) \) and \( y(t) \) draw a path in the coordinate plane. Observing the behavior of \( x \) and \( y \) as \( t \) varies helps in accurately plotting the curve on a graph.
To graph the function \( x = \csc t \) and \( y = \cot t \), observe both:

• As \( t \) ranges from \(-\pi \) to \(0 \), \( \csc t \) influences \( x \) to vary from \(-\infty \) to values close to \(-1 \). This indicates the path traversing from left to right.
• Simultaneously, \( \cot t \) affects \( y \) values to remain non-positive, indicating the journey through the fourth quadrant.

These graphing techniques are fundamental to visually understanding how parameters affect the shape and direction of a curve.

Coordinate Geometry

Coordinate Geometry, or analytic geometry, connects algebra and geometry to describe the positions of points, lines, and figures in space. Using coordinates, we can express relations and quantify geometric properties.
When reworking parametric equations into Cartesian coordinates, as in transforming \( x = \csc t \) and \( y = \cot t \) into \( y = \sqrt{x^2 - 1} \), this field helps describe geometrical shapes and their characteristics.

• Useful for calculating distances and midpoints along a graph.
• Helps in deriving equations of geometrical figures, such as lines or circles, in a Cartesian plane.
• Enables easier interpretation through graphical representation.

Such transformations allow us to appreciate the underlying geometrical structure and ensure precise plotting and analysis.