Problem 54 - Thomas Calculus

Question

Replace the Cartesian equations in Exercises \(53-66\) with equivalent polar equations. $$y=1$$

Step-by-Step Solution

Answer

The polar equation is \(r = \csc\theta\).

Step 1: Understand Cartesian and Polar Coordinates

Cartesian coordinates are typically written as \(x, y\) and describe a point's position on the plane relative to two perpendicular axes. Polar coordinates express a point using a distance \(r\) from the origin, and an angle \(\theta\) from the positive x-axis. Formulas to convert between these systems include \(x = r\cos\theta\) and \(y = r\sin\theta\).

Step 2: Express the Cartesian Equation using Polar Formulas

The given Cartesian equation is \(y = 1\). We know that in polar coordinates, \(y\) is equal to \(r\sin\theta\). Substitute \(y\) with \(r\sin\theta\) in the equation: \r\sin\theta = 1\.

Step 3: Solve for r in terms of \(\theta\)

The polar conversion result is \(r\sin\theta = 1\). Solve this equation for \(r\) by dividing both sides by \(\sin\theta\), assuming \(\theta eq k\pi\) (since \(\sin\theta\) would be zero): \r = \frac{1}{\sin\theta} = \csc\theta\.

Step 4: Write the Final Polar Equation

The equivalent polar equation for \(y = 1\) is \(r = \csc\theta\). This represents all points where the vertical distance to the x-axis (y = 1) is consistent, expressed in terms of polar coordinates.

Key Concepts

Cartesian Coordinates

To understand the concept of Cartesian coordinates, imagine a grid with two lines crossing each other at the center, similar to a 'plus' sign. These lines, called axes, are the x-axis (horizontal) and y-axis (vertical). Points on this grid are identified by their position in relation to these axes.

For example, the point (3, 2) on the grid means you move 3 units right along the x-axis and 2 units up along the y-axis from the center or origin. This is how Cartesian coordinates are used to precisely define points in a 2D space. The ordered pair (x, y) gives all the necessary information about a point's location.

Understanding Cartesian coordinates is essential in mathematics because it forms the basis for graphing equations and understanding spatial relationships.

• The x-coordinate tells us how far the point is horizontally from the origin.
• The y-coordinate tells us how far the point is vertically from the origin.
• Both figures combine to give the point's exact location.

If you can grasp this system, you'll easily navigate more advanced concepts such as coordinate conversion and equations.

Coordinate Conversion

Coordinate conversion is the process of transforming points from one system, like Cartesian, to another, such as polar coordinates.

Different systems serve different purposes, and sometimes it's easier to solve problems in one coordinate system over another. That's why conversion is essential.

To convert from Cartesian (x, y) to polar (r, \( \theta \)) coordinates, use the formulas:

• \( r = \sqrt{x^2 + y^2} \): This calculates the distance from the origin to the point, essentially the radius.
• \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \): This gives the angle formed with the positive x-axis.

Using these formulas, you can easily switch between systems. The reverse conversion is equally straightforward using the equations:

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

Knowing how to perform coordinate conversion allows you to express equations and solve problems in the most efficient way possible. This is particularly helpful in fields like physics, engineering, and computer graphics.

Polar Equations

Polar equations use polar coordinates to describe points and curves in a plane. Unlike Cartesian equations that focus on horizontal and vertical distances, polar equations express relationships based on a point's distance from the origin and the angle it forms with the positive x-axis.

For instance, consider the Cartesian equation \( y = 1 \), which describes a horizontal line. When converted to polar coordinates, the equation becomes \( r = \csc\theta \). This equation means the distance from the origin is inversely related to the angle's sine value.

Polar equations often simplify the representation of curves like spirals and circles, which might appear complex when expressed in Cartesian coordinates.

• Circular paths are often more naturally represented in polar form.
• Understanding these equations helps solve many practical problems, including those involving cyclic and radial patterns.

By mastering polar equations, you improve your ability to work with geometric and spatial problems, making it easier to visualize and manipulate forms in mathematical space.