Problem 32 - Thomas Calculus

Question

Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian equations. Then describe or identify the graph. $$r=-3 \sec \theta$$

Step-by-Step Solution

Answer

The Cartesian equation is \(x = -3\). It's a vertical line through \(x = -3\).

Step 1: Recall the Polar to Cartesian Conversion Formulas

Recall that in polar coordinates, \(r\) is the radius, and \(\theta\) is the angle. The conversion formulas to change from polar to Cartesian coordinates are \(x = r \cos \theta\) and \(y = r \sin \theta\). Also, remember the identity \(\sec \theta = \frac{1}{\cos \theta}\).

Step 2: Substitute the Polar Equation into the Identity

Given the equation \(r = -3 \sec \theta\), substitute \(\sec \theta = \frac{1}{\cos \theta}\). This gives \(r = \frac{-3}{\cos \theta}\). Multiply both sides by \(\cos \theta\) to clear the fraction: \(r \cos \theta = -3\).

Step 3: Convert the Polar Equation to Cartesian Form

Use the conversion formula \(x = r \cos \theta\) from Step 1. Plugging in, we get \(x = -3\).

Step 4: Describe the Graph

The equation \(x = -3\) represents a vertical line in the Cartesian coordinate system that crosses the x-axis at \(-3\).

Key Concepts

Polar Coordinates

Polar coordinates are a way of describing a point in a plane. They consist of two components: the radius \( r \) and the angle \( \theta \). The radius is the distance from the point to the origin (0,0), while the angle \( \theta \) is measured in radians or degrees from the positive x-axis. This system is particularly useful when dealing with circular or rotational problems because it directly relates to the geometry of circles.

In a polar coordinate system, every point is described by how far it is from the origin and the angle it makes with the central reference line (often the x-axis). This setup assists in easily representing the locations and movements of objects around a center point. Polar coordinates are not unique, meaning that there can be multiple representations of the same point, depending on how many rotations are considered in adding or subtracting full circle equivalents \((360^{\circ} \text{ or } 2\pi)\).

• The formula \( r = \pm\sqrt{x^2+y^2} \) can be used to find the distance from the origin.
• The angle \( \theta = \arctan\left(\frac{y}{x}\right) \) can help determine the rotation from the positive x-axis to the point.

Cartesian Coordinates

Cartesian coordinates express the location of a point using an x and a y value in a two-dimensional space. This approach creates a rectangular coordinate system, where each point is represented by a pair \((x, y)\). The x-coordinate refers to the horizontal position, and the y-coordinate refers to the vertical position of the point. Cartesian coordinates are a foundation of geometry and algebra because they allow for geometric reasoning in a straightforward way.

Converting from polar to Cartesian coordinates involves using the relationships:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

These allow you to translate the circular descriptions of polar coordinates into the linear descriptions of Cartesian coordinates. This conversion is crucial when you need to perform calculations or draw graphs in Cartesian planes, as it clarifies how angles and distances relate to horizontal and vertical positions.

The Cartesian coordinate system is commonly used in analytical geometry, algebra, and calculus, providing a versatile tool for graphing equations and functions as it accommodates all kinds of transformations, symmetries, and scaling techniques.

Graphing Equations

Graphing equations often begins with understanding how to plot them in a coordinate system. When dealing with polar equations, converting them to Cartesian form can help plot these graphs effectively on a typical x-y grid. Once converted, it becomes easier to visualize what the equation represents.

For the equation \( r = -3 \sec \theta \), when converted to Cartesian coordinates, it simplifies to \( x = -3 \), representing a simple vertical line crossing the x-axis at -3. This conversion process shows how the original equation in polar coordinates can manifest in a straightforward geometric figure on the Cartesian plane.

To successfully graph equations:

• Identify if it's easier to graph an equation directly in polar form or by converting to Cartesian.
• Understand key features of the equation, such as intercepts and symmetries.
• Use plotting tools or graph paper to draw accurate representations.

These strategies facilitate recognition of patterns and relationships inherent in mathematical equations, aiding in the comprehension and solving of complex geometric problems.