Question

Describe the graph of each polar equation. Confirm each description by converting into a rectangular equation. $$ r=\csc \theta $$

Short answer

The graph is a horizontal line at \( y = 1 \).

Step-by-step solution

Understand Polar Equation

The given polar equation is \( r = \csc \theta \). In polar coordinates, the variable \( r \) denotes the radial distance from the origin, and \( \theta \) is the angle from the positive x-axis.

Convert Cosecant to Sine

The cosecant function is the reciprocal of the sine function, so we can express the equation as \( r = \frac{1}{\sin \theta} \).

Cross-Multiply

To eliminate the fraction, we can multiply both sides by \( \sin \theta \): \( r \cdot \sin \theta = 1 \).

Use Polar-to-Rectangular Conversion

Know that in polar to rectangular conversion: \( x = r \cos \theta \) and \( y = r \sin \theta \). Here we focus on \( \sin \theta = \frac{y}{r} \). Therefore, \( r \cdot \sin \theta = y \).

Rewrite Equation in Rectangular Form

Substitute from Step 4 \( y = 1 \) into the earlier equation. This shows that the rectangular equation of the given polar graph is a horizontal line where the y-coordinate is always 1.

Key concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are used to define a point in the plane by two numerical values: the x-coordinate (horizontal axis) and the y-coordinate (vertical axis). These two numbers together specify a point's position in the two-dimensional space.

Some key points to understand about rectangular coordinates include:

• The x-coordinate represents the horizontal distance from the origin, which is the point (0,0).
• The y-coordinate represents the vertical distance from the origin.
• Points are plotted by moving x units along the x-axis and y units along the y-axis.

For example, the coordinate (3,4) means we start at the origin, move 3 units along the x-axis, and then move 4 units up along the y-axis.

Rectangular coordinates are widely used in everyday mathematics and are essential for graphing equations in planar geometry. They make it easier to understand the concept of slope, distance, and various geometric shapes like lines and circles.

Polar to Rectangular Conversion

Converting from polar to rectangular coordinates involves transforming radial and angular measurements into x and y coordinates. This is particularly useful when you want to analyze or graph polar equations using a more familiar Cartesian grid.

To convert, you utilize the relationships:

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

These formulas arise from a right triangle where \( r \) is the hypotenuse. The angle \( \theta \) determines the direction in which the hypotenuse extends.

In our specific conversion story for the polar equation \( r = \csc \theta \), we redefined the equation in terms of sine because \( \csc \theta = \frac{1}{\sin \theta} \). By substituting directly into the conversion formula for \( y \), we revealed that this equation simplifies to a horizontal line: \( y = 1 \).

Understanding these conversions helps bridge the gap between two fundamental methods of describing a point's location and is a valuable skill for interpreting complex graphs in various mathematical applications.

Graphing Polar Equations

Graphing polar equations involves plotting points based on their distance and angle from the origin. This approach differs from Cartesian graphing because it focuses on circular plots rather than linear grids. Polar coordinates are particularly effective for graphs that exhibit symmetrical or rotational patterns.

Key aspects of graphing in polar coordinates include:

• The radial coordinate \( r \) states how far from the origin the point lies.
• The angular coordinate \( \theta \) specifies the angle formed with the positive x-axis.

To graph a polar equation like \( r = \csc \theta \), you start by determining the shape from the equation structure. Here, because \( \csc \theta \) is undefined at \( \theta = 0 \) \( \text{and multiples of } \pi \), the graph shows unique behavior around these angles.

After the polar-to-rectangular conversion, where we found that the y-coordinate consistently equals 1, plotting becomes straightforward: the graph is a horizontal line crossing the Cartesian plane at y = 1. This visualization step helps in understanding how polar descriptions transform into more familiar rectangular plots, aiding in comprehensive analysis of mathematical functions.