Problem 166 - Calculus Volume 3

Question

Convert the polar equation to rectangular form and sketch its graph. $$ r=\cot \theta \csc \theta $$

Step-by-Step Solution

Answer

The rectangular form is \( y^2 = x \), which is a parabola opening to the right.

Step 1: Recall Polar to Rectangular Coordinate Conversion Formulas

To convert the polar equation to rectangular form, use the standard conversions: \[ x = r \cos\theta, \quad y = r \sin\theta, \quad r^2 = x^2 + y^2, \quad \text{and} \quad \tan\theta = \frac{y}{x}. \]

Step 2: Express Cotangent and Cosecant in Terms of Sine and Cosine

Recognize that \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \). Hence, the polar equation becomes \[ r = \cot \theta \csc \theta = \frac{\cos \theta}{\sin \theta} \cdot \frac{1}{\sin \theta} = \frac{\cos \theta}{\sin^2 \theta}. \]

Step 3: Substitute Polar to Rectangular Conversion

Substitute the known definitions into the equation by recalling that \( \sin \theta = \frac{y}{r} \) and \( \cos \theta = \frac{x}{r} \). Therefore, \[ r = \frac{\frac{x}{r}}{\left(\frac{y}{r}\right)^2} = \frac{x}{y^2}. \]

Step 4: Multiply Through by \( y^2 \) to Eliminate Fraction

Multiply every term of the equation by \( y^2 \) to eliminate the division, resulting in the rectangular equation: \[ y^2 = x. \]

Step 5: Identify the Type of Graph

The equation \( y^2 = x \) represents a parabola opening to the right. Use this insight to sketch the graph.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are based on a pair of perpendicular axes, typically labeled x and y. These coordinates represent the standard way to describe any point in a plane. Each point on a plane can be described using this format:

• x - the horizontal distance from the origin (0,0).
• y - the vertical distance from the origin (0,0).

Using these coordinates, equations can describe lines, curves, and other shapes within the plane. Rectangular form is especially intuitive for graphing and mathematical operations because it directly relates to the common xy-grid used in most classes and applications.In the exercise, once converted, the equation becomes \( y^2 = x \).This form is much simpler to understand and graph using rectangular coordinates.

Polar Coordinates

Polar coordinates offer a different way of locating a point on a plane. Instead of using the traditional x and y axes intersections, polar coordinates use:

• r - the distance from the origin to the point.
• \(\theta\) - the angle from the positive x-axis to the radius line, measured in radians or degrees.

This system is especially useful for curves and figures that are radii or arcs-centric, such as circles and spirals.These coordinates allow us to write equations that may seem complicated in rectangular form but are quite simple in polar form. For example, a circle centered at the origin is simply \( r = a \), a constant. In the given problem, the polar equation \( r = \cot \theta \csc \theta \) presents a unique case that becomes more straightforward after conversion to rectangular coordinates.

Converting Equations

Converting equations from polar to rectangular forms, or vice versa, can be quite insightful and sometimes necessary. The conversion is accomplished using several key relationships:

• \( x = r \cos\theta \), \( y = r \sin\theta \)
• \( r^2 = x^2 + y^2 \)
• \( \tan\theta = \frac{y}{x} \)

These transformations allow us to change the format of complex equations to make them easier to work with or to better understand the graph of the expression.In our exercise, we used the identities for cotangent and cosecant to simplify and transform the initial polar equation into a rectangular format by substituting \( \sin\theta \) and \( \cos\theta \)

Parabola Graphing

Parabolas are one of the first quadratic shapes many students learn to graph. In the context of analytical geometry, parabolas open either vertically or horizontally. The general forms in rectangular coordinates are:

• \( y^2 = 4ax \) (parabola opens to the left or right)
• \( x^2 = 4ay \) (parabola opens up or down)

The parabola in this problem, achieved after converting the equation, is \( y^2 = x \), which opens to the right. Graphing a parabola involves plotting points that satisfy the equation and sketching the U-shaped curve that passes through them. In this case, as x increases, y corresponds to positive and negative square roots of x, reflecting the symmetry of the parabola about the x-axis.