Problem 47 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the polar equation. Describe your viewing window. $$r=2 \csc \theta+6$$

Step-by-Step Solution

Answer

The graph of the polar equation \( r=2 \csc \theta + 6 \) is plotted using a graphing utility with the viewing window adjusted such that \( r \) ranges from -10 to 10 and \( \theta \) varies between 0 and \( 2\pi \) radians (or 0 to 360 degrees).

Step 1: Input the Polar Equation

Using a graphing tool that supports polar coordinates, input the polar equation \( r=2 \csc \theta + 6 \) in the provided field.

Step 2: Adjust the Viewing Window

Adjust the viewing window to clearly visualize the plot of the polar equation. A suitable range for \( r \) would be from -10 to 10, and \( \theta \) should vary between 0 and \( 2\pi \) radians (or 0 to 360 degrees).

Step 3: Generate the Graph

Hit the graph button or equivalent action to generate the plot of the equation in the adjusted viewing window.

Step 4: Describe the Viewing Window

The window where the graph appears is referred to as the viewing window. It has been adjusted such that \( r \) ranges from -10 to 10, and \( \theta \) varies between 0 and \( 2\pi \) radians (or 0 to 360 degrees). Describe what you see in the viewing window with respect to the plot of the equation.

Key Concepts

Polar Coordinates

Polar coordinates are an alternative to the Cartesian coordinate system used to represent mathematical concepts and physical locations. In polar coordinates, each point on a plane is determined by a distance from a reference point, called the pole (similar to the origin in Cartesian coordinates), and an angle from a reference direction, usually the positive x-axis.

The notation \( r, \theta \) represents a polar coordinate, where \( r \) is the radial distance from the pole and \( \theta \) is the angular coordinate, typically measured in radians. Polar coordinates are particularly useful for dealing with equations involving angles and lengths, and they are commonly used in fields such as physics, engineering, and navigation.

Cosecant Function

The cosecant function, denoted as \( \text{csc} \) or sometimes \( \text{sec}^{-1} \), is one of the six trigonometric functions and it is the reciprocal of the sine function. It's defined as \( \text{csc}(\theta) = \frac{1}{\sin(\theta)} \) for all angles \( \theta \) where \( \sin(\theta) eq 0 \).

In a polar equation such as \( r=2 \text{csc}(\theta) + 6 \), the cosecant function's effect can be visualized as a variation in the radial distance \( r \) depending on the angle \( \theta \) — this results in a curve that stretches outwards wherever the sine of \( \theta \) is small, and comes closer in where the sine is large. Graphing this function can show interesting patterns, like loops or petals that are commonly seen in polar graphs generated by trigonometric functions.

Graphing Utility

A graphing utility is an essential tool in mathematics for visualizing equations and functions. This can be accomplished through a graphing calculator, computer software, or online applications that plot graphs based on given inputs. When working with polar coordinates, it is important that the graphing utility supports polar equations, as they require the plotting of points using the radius and angle rather than pure horizontal and vertical axes.

Many utilities offer interactive features, allowing for real-time adjustments and immediate visual feedback. This is particularly helpful when exploring mathematical concepts and understanding the behavior of different functions, which is challenging to conceptualize without a visual aid.

Viewing Window Adjustment

Adjusting the viewing window is a crucial step when graphing equations, as it ensures that the essential parts of the graph are visible and allows for a better interpretation of the resulting plot. The viewing window defines the range of values for both the radial and angular coordinates that the graphing utility will display.

In the context of the provided exercise, setting a viewing window from -10 to 10 for \( r \) and from 0 to \( 2\pi \) (or 0 to 360 degrees) for \( \theta \) gives a complete one-period graph of the cosecant function in polar coordinates. This adjustment lets us view the full behavior of the polar equation without distortion or incomplete sections, providing more insight into the characteristics of the graph. Enhancing the understanding of this adjustment can greatly assist students in successfully interpreting and graphing complex functions.