91Ó°ÊÓ

An inequality is a mathematical statement that defines the relationship between two values that are not equal. In this exercise, the inequality is given by \[0 \leq r^2 \leq \cos \theta\]. This inequality tells us that the square of the radius, \(r^2\), must be non-negative and not greater than \(\cos \theta\).
This is important, as it defines the range of \(r\) in terms of \(\theta\).

In essence, the inequality splits into two parts:

• \(r^2 \geq 0\), which is always true since squaring any real number results in a non-negative value.
• \(r^2 \leq \cos \theta\), which imposes the condition that \(r\) can only equate to a square root of a non-negative cosine value.

This restricts our consideration to angles where \(\cos \theta\) is non-negative, ensuring that \(r\) is real and positive, or zero.

The polar plot is a graphical representation where each point on the plane is defined by a distance from a reference point and an angle from a reference direction. In the context of our inequality \[0 \leq r \leq \sqrt{\cos \theta}\], a polar plot helps visualize the region satisfying these conditions. In polar coordinates, the values of \(r\) vary based on \(\theta\), giving us a curve or region instead of a single line or shape in Cartesian coordinates.

Choosing \(r\) values from 0 to \(\sqrt{\cos \theta}\) as \(\theta\) varies from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), we can sketch these points in the polar plane. The reference direction is usually the positive x-axis.

• The angle \(\theta = 0\) corresponds to the positive x-axis direction.
• \(r = \sqrt{\cos \theta}\) gives us the distance from the origin.
• As we increase \(\theta\), the radius changes, creating a dynamic shape in the plot.

The result is a symmetrical figure around the y-axis in the plane.

Trigonometric functions relate the angles of a triangle to the lengths of its sides. In our exercise, the cosine function is key. \(\cos \theta\) determines the maximum allowable value of \(r^2\), limiting our polar plot area.

Understanding cosine helps us know when the inequality holds:

• \(\cos \theta\) is positive in the first and fourth quadrants (i.e., angles between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\)).
• At \(\theta = 0\), \(\cos \theta\) reaches its peak value of 1, making \(r = 1\) the longest radial distance in our plot.

Thus, trigonometric functions, particularly cosine, are pivotal in determining the configuration of the sketched region.

Region sketching in polar coordinates involves plotting points that satisfy given conditions, then interpreting these points as a complete shape or region. For inequality \[0 \leq r \leq \sqrt{\cos \theta}\], region sketching starts by identifying acceptable \(r\) values for each \(\theta\).

To accomplish this:

• Fix \(\theta\) within the allowable range (\(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\)) where \(\cos \theta\) is non-negative.
• Calculate corresponding \(r\) values using \(r = \sqrt{\cos \theta}\).
• Plot these points on the polar plane, joining them to form the region satisfying the inequality.

This constructed area is symmetrical, resembling a cardioid but bounded by the cosine's influence.
At \(\theta = 0\), the region hits its maximum width, tapering as \(\theta\) nears limits to \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). This approach transforms equations into a visual understanding, anchoring abstract math in tangible geometry.