Problem 50 - Thomas Calculus

Question

Replace the Cartesian equations in Exercises \(49-62\) by equivalent polar equations. $$ y=1 $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Cartesian Equation

The given Cartesian equation is \( y = 1 \). This is a horizontal line that crosses the y-axis at 1 in the Cartesian coordinate system.

Step 2: Use Polar Coordinate Definitions

Recall that in polar coordinates, any point can be represented as \( (r, \theta) \), where \( r \) is the radius (or distance from the origin) and \( \theta \) is the angle from the positive x-axis. The conversion formulas from Cartesian coordinates are: \( x = r \cos \theta \) and \( y = r \sin \theta \).

Step 3: Substitute for y in the Polar Form

Substitute \( y = r \sin \theta \) into the Cartesian equation. The equation \( y = 1 \) becomes \( r \sin \theta = 1 \).

Step 4: Solve for r in terms of θ

Isolate \( r \) in the equation \( r \sin \theta = 1 \) by dividing both sides by \( \sin \theta \). This gives us \( r = \frac{1}{\sin \theta} = \csc \theta \).

Step 5: Write the Polar Equation

The equivalent polar equation is \( r = \csc \theta \), which is the representation of the line \( y = 1 \) in polar coordinates.

Key Concepts

Cartesian coordinates

Cartesian coordinates are a way to specify the location of a point in a plane. These coordinates use two perpendicular lines called axes: the x-axis (horizontal) and the y-axis (vertical). Any point in this plane can be described with an ordered pair

• The first number, known as the 'x-coordinate', tells us how far the point is horizontally from the origin (0,0).
• The second number, the 'y-coordinate', indicates the point's vertical distance from the origin.

For instance, in the equation \( y = 1 \), this represents a horizontal line crossing the y-axis at the point (0,1), meaning every point on this line has a y-coordinate of 1. This is the beauty of Cartesian coordinates—they make it easy to visualize and graph relationships between variables.

coordinate conversion

Coordinate conversion is a crucial mathematical process when switching between different types of coordinate systems. Cartesian and polar coordinates are both useful, but sometimes one is more suitable than the other for a particular task. To convert from Cartesian to polar coordinates, you use the following formulas:

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

In these formulas, \( r \) represents the distance from the origin to the point, and \( \theta \) is the angle the line makes with the positive x-axis.For the Cartesian equation \( y = 1 \), you use these conversion formulas. Substitute \( y = r \sin \theta \) into this equation to get \( r \sin \theta = 1 \). Solving for \( r \), you find \( r = \frac{1}{\sin \theta} = \csc \theta \). Now, the equation in polar form is ready. This conversion process helps in understanding complex figures and equations by shifting the perspective of observation.

polar equations

Polar equations express relationships using polar coordinates. They are incredibly effective for dealing with scenarios where the location of points is based on direction and magnitude rather than x-y position. Polar coordinates represent each point as \( (r, \theta) \):

• \( r \) is the radius, or how far the point is from the origin.
• \( \theta \) is the angle formed with the positive x-axis.

When converting \( y = 1 \) into a polar equation, you use \( r \sin \theta \) to replace \( y \). The polar form turns into \( r = \csc \theta \) which describes the same line depicted by the equation \( y = 1 \). By representing lines and curves in polar form, especially if they originally present challenges when described in Cartesian terms, you can easily analyze them, especially for curves that have symmetry around a point rather than around a line.