Radio Towers and Broadcasting Patterns Radio stations do not...

Radio Towers and Broadcasting Patterns Radio stations do not always broadcast in all directions with the same intensity. To avoid interference with an existing station to the north, a new station may be licensed to broadcast only east and west. To create an east-west signal, two radio towers are sometimes used. See the figure. Locations where the radio signal is received correspond to the interior of the curve $$ r^{2}=40,000 \cos 2 \theta $$ where the polar axis (or positive $x$ -axis) points east. (a) Graph $r^{2}=40,000 \cos 2 \theta$ for $0^{\circ} \leq \theta \leq 360^{\circ},$ where distances are in miles. Assuming the radio towers are located near the pole, use the graph to describe the regions where the signal can be received and where the signal cannot be received. (b) Suppose a radio signal pattern is given by the following equation. Graph this pattern and interpret the results. $$ r^{2}=22,500 \sin 2 \theta $$ (GRAPH CANT COPY)

Radio Towers and Broadcasting Patterns Radio stations do not always broadcast in all directions with the same intensity. To avoid interference with an existing station to the north, a new station may be licensed to broadcast only east and west. To create an east-west signal, two radio towers are sometimes used. See the figure. Locations where the radio signal is received correspond to the interior of the curve
$$
r^{2}=40,000 \cos 2 \theta
$$
where the polar axis (or positive $x$ -axis) points east.
(a) Graph $r^{2}=40,000 \cos 2 \theta$ for $0^{\circ} \leq \theta \leq 360^{\circ},$ where distances are in miles. Assuming the radio towers are located near the pole, use the graph to describe the regions where the signal can be received and where the signal cannot be received.
(b) Suppose a radio signal pattern is given by the following equation. Graph this pattern and interpret the results.
$$
r^{2}=22,500 \sin 2 \theta
$$
(GRAPH CANT COPY)

Key Concepts

Lemniscate and Rose Curves

Equations involving terms like cos2? or sin2? often yield curves that have lobed or figure?8 shapes, such as lemniscates or rose curves. Grasping the general characteristics of these curves helps in recognizing them and analyzing their properties even before plotting, which is important in understanding patterns with directional bias.

Interpreting Bounded Regions

This involves understanding which areas of the plane are enclosed by the curve defined by a polar equation. Such interpretation is crucial when the equation represents a boundary of a region, as in applications where the existence of a physical phenomenon (like a signal) is confined to the interior or exterior of the curve.

Polar Coordinates

This concept refers to a system for representing points in a plane using a distance from a fixed origin and an angle from a fixed direction. It is particularly useful for describing curves that are more naturally expressed in terms of angles and radii rather than x and y coordinates.

Graphing Polar Equations

Graphing polar equations involves plotting points by calculating the radius for various angles and then mapping these points in the polar plane. Understanding how to handle negative r values, periodicity, and the behavior of trigonometric functions is key to accurately sketching the graph.

Symmetry in Polar Graphs

Many polar graphs exhibit symmetrical properties with respect to the polar axis, the line ? = ?/2, or the pole itself. Recognizing these symmetries simplifies the graphing process by reducing the range of angles that need to be plotted and aids in interpreting the overall behavior of the curve.

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