Problem 54 - Precalculus Mathematics for Calculus

Question

Convert the polar equation to rectangular coordinates. $$r=2 \csc \theta$$

Step-by-Step Solution

Answer

The rectangular equation is \( y = 2 \).

Step 1: Understand the relationship

The given polar equation is \( r = 2 \csc \theta \). To convert it to rectangular coordinates, we need to use the relationships between polar and rectangular coordinates: \( x = r \cos \theta \), \( y = r \sin \theta \), and \( r^2 = x^2 + y^2 \). Also, \( \sin \theta = \frac{y}{r} \) and \( \csc \theta = \frac{1}{\sin \theta} \).

Step 2: Substitute \( \csc \theta \)

Substitute \( \csc \theta = \frac{1}{\sin \theta} \) into the given equation, thus: \[ r = 2 \csc \theta = \frac{2}{\sin \theta} \]

Step 3: Solve for \( \sin \theta \)

Rearrange the equation to solve for \( \sin \theta \): \[ r \sin \theta = 2 \] Using the relationship \( \sin \theta = \frac{y}{r} \), substitute it into the equation to get: \[ r \cdot \frac{y}{r} = 2 \]Simplify this equation to obtain \( y = 2 \).

Step 4: Write the Rectangular Equation

The final equation in rectangular coordinates does not involve \( x \) but shows a consistent value for \( y \): \[ y = 2 \]Thus, the equation in rectangular coordinates is simply a horizontal line at \( y = 2 \).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also widely known as Cartesian coordinates, are a system that uses two perpendicular axes to specify the location of a point on a plane. The two axes are conventionally labeled as the x-axis and the y-axis. Any point on this plane is defined by a pair of numerical coordinates:

• x-coordinate: Represents the horizontal distance from the origin.
• y-coordinate: Represents the vertical distance from the origin.

Together, these coordinates \(x, y\) uniquely identify a point in the plane. This system is immensely useful due to its simplicity and the ability to easily interpret geometric shapes and graphs. One common feature is that lines can be easily represented with linear equations, such as the rectangular equation \( y = 2 \), which describes a horizontal line that remains constant along the y-axis and parallel to the x-axis.

Polar Coordinates

Polar coordinates provide an alternative way of representing points in a plane using a different framework. Rather than being based on perpendicular lines, polar coordinates use a circle-centric approach:

• r (radius): The distance from the origin to the point.
• \(\theta\) (theta): The angle formed with the positive x-axis, typically measured in radians.

In this system, each point is identified with a pair \(r, \theta\). The polar coordinate system is especially handy in situations where the geometry of the problem is naturally circular, such as in cases of circular motion. Unlike the rectangular system, the same point can have different polar coordinates due to the cyclical nature of angles.

Coordinate Conversion

Converting between polar and rectangular coordinates is a vital skill for solving many mathematical problems. This conversion relies on foundational trigonometric relationships:

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

In the provided exercise, you began with a polar equation \(r = 2 \csc \theta\) which was converted to the rectangular format. Using polar identities like \(\csc \theta = \frac{1}{\sin \theta}\), the equation was transformed and simplified based on the identity \(\sin \theta = \frac{y}{r}\) to ultimately result in the equation \(y = 2\). This conversion process is invaluable in allowing different mathematical properties and tools to be used effectively in problem-solving, depending on the situation.