Problem 20 - Calculus Early Transcendentals

Question

\(15-20\) Identify the curve by finding a Cartesian equation for the curve. $$r=\tan \theta \sec \theta$$

Step-by-Step Solution

Answer

The Cartesian equation is \(x^2 = y\), representing a parabola.

Step 1: Recall Polar to Cartesian Conversion Formulas

To convert the polar equation into a Cartesian form, recall the formulas: \(x = r \cos \theta\) and \(y = r \sin \theta\). Also, remember \(\tan \theta = \frac{y}{x}\) and \(\sec \theta = \frac{1}{\cos \theta}\).

Step 2: Incorporate Given Polar Equation

Given the equation \(r = \tan \theta \sec \theta\), express \(\tan \theta\) and \(\sec \theta\) in terms of \(x\) and \(y\): \(\tan \theta = \frac{y}{x}\) and \(\sec \theta = \frac{1}{\cos \theta} = \frac{r}{x}\). Substitute these into the equation: \(r = \frac{y}{x} \cdot \frac{r}{x}\).

Step 3: Simplify the Equation

Simplify the equation from Step 2: \(r = \frac{yr}{x^2}\). Multiply both sides of the equation by \(x^2\) to eliminate the fraction: \(rx^2 = yr\).

Step 4: Eliminate r from the Equation

Since \(r = \sqrt{x^2 + y^2}\) in polar coordinates, substitute \(r\) with \(\sqrt{x^2 + y^2}\) from the equation: \(\sqrt{x^2 + y^2}x^2 = y\sqrt{x^2 + y^2}\).

Step 5: Further Simplify

Divide both sides of the equation by \(\sqrt{x^2 + y^2}\) (assuming \(r eq 0\)) to get \(x^2 = y\).

Step 6: Write the Cartesian Equation

The simplified Cartesian equation is \(x^2 = y\), which represents a parabola opening upwards on the xy-plane.

Key Concepts

Polar Coordinates

Polar coordinates are a flexible way to represent points in a plane through a distance and angle rather than traditional x and y coordinates. Often denoted as \((r, \theta)\), where:

• \(r\) is the radial distance from the origin (0,0) to the point.
• \(\theta\) is the angle measured from the positive x-axis counterclockwise to the point.

Using this system allows for a more natural description of circular or rotational movement, which can be quite beneficial for various areas such as physics and engineering.
For example, moving around a circle, it's easier to specify an angle and radius than Cartesian coordinates. However, in some cases, it becomes necessary to convert these coordinates into the more commonly used Cartesian form.

Cartesian Coordinates

Cartesian coordinates use two perpendicular axes to describe a point in the plane by their horizontal and vertical components, called the x-coordinate and y-coordinate, respectively. Imagine superimposing a grid onto a piece of paper; each grid square represents a unit of measurement.

• The horizontal axis is typically labeled as the x-axis.
• The vertical axis is known as the y-axis.
• A point is denoted as \((x, y)\).

Cartesian coordinates are a standard method for plotting data in mathematics and science, making it convenient for calculations and visualizations of relationships between variables.
Converting polar coordinates into Cartesian coordinates can facilitate the solving of geometric problems or graphing functions. With transformation formulas such as \(x = r \cos \theta\) and \(y = r \sin \theta\), a polar equation can be expressed in Cartesian form for further mathematical analysis.

Equation Transformation

Equation transformation often involves re-casting equations from one form to another to simplify operations or gain a new perspective. In polar and Cartesian conversions, this translation is essential for understanding and working with curves and functions across different systems.
To transform the polar equation \(r = \tan \theta \sec \theta\) into a Cartesian equation, we apply several trigonometric identities and transformations:

• Recall \(\tan \theta = \frac{y}{x}\) and \(\sec \theta = \frac{1}{\cos \theta}\), resulting in \(\sec \theta = \frac{r}{x}\).
• Substitute these into the equation to express all components in terms of \(x\) and \(y\).
• The equation simplifies and results in \(x^2 = y\), illustrating how radical expressions are eliminated when possible.

Equation transformation like this allows for the determination of graph shapes and characteristics in a more universally readable format.

Parabola

A parabola is a curve where each point is equidistant from a fixed point, known as the focus, and a fixed line, called the directrix. The equation \(x^2 = y\) represents a parabola in the Cartesian plane, opening upwards.

• It is symmetric with respect to the y-axis due to the squared term \(x^2\).
• Its vertex is located at the origin (0,0).

Parabolas are a common sight in mathematics and physics due to their unique reflective properties and appearance in graphing quadratic functions. Understanding the transformation from polar to Cartesian coordinates highlights the consistent nature of geometric figures across different coordinate systems.
These attributes make analyzing and sketching the graph manageable and useful for various applications, including projectile motion and optimizing solutions.