Problem 49 - College Trigonometry

Question

In Exercises \(41-50\), use set-builder notation to describe the polar region. Assume that the region contains its bounding curves. The region which lies inside of the circle \(r=3 \cos (\theta)\) but outside of the circle \(r=\sin (\theta)\)

Step-by-Step Solution

Answer

The region is \( \{(r, \theta) \mid \sin \theta \leq r \leq 3 \cos \theta, 0 \leq \theta < 2\pi \} \).

Step 1: Define the First Circle Equation

First, we need to understand what the equation \(r = 3 \cos \theta\) represents. This is the equation of a polar circle with its center at \( (3/2, 0) \) in Cartesian coordinates and a radius \(3/2\). It represents all points \((r, \theta)\) such that \( r \leq 3 \cos \theta \).

Step 2: Define the Second Circle Equation

Next, examine the equation \(r = \sin \theta\). This is another polar circle centered at \((0, 1/2)\) with a radius of \(1/2\). It represents all points \((r, \theta)\) such that \( r \geq \sin \theta\).

Step 3: Combine the Circle Conditions

We need to describe the region inside the first circle but outside the second circle. This region is defined by the inequalities: \(\sin \theta \leq r \leq 3 \cos \theta\).

Step 4: Express in Set-Builder Notation

Now we express this region using set-builder notation. The set-builder notation for points in this region is: \[ \{(r, \theta) \mid \sin \theta \leq r \leq 3 \cos \theta, 0 \leq \theta < 2\pi \} \]. This denotes the set of all points \((r, \theta)\) which satisfy both conditions.

Key Concepts

Polar Coordinates

Polar coordinates represent a way of describing the location of a point in a plane using two values: the radial distance from a reference point (often the origin) and the angle measured from a reference direction (typically the positive x-axis). This system differs from the Cartesian coordinate system, which uses a pair of perpendicular axes to pinpoint positions.

In polar coordinates:

• The radial distance, often denoted as \(r\), gives how far the point is from the origin.
• The angle, \(\theta\), describes the counterclockwise direction from the positive x-axis to the point.

Polar coordinates are particularly useful when dealing with circular or angular forms, as they can simplify expressions and provide more intuitive insights into rotational problems. Converting between polar and Cartesian coordinates can be done using the following formulas:

• From polar to Cartesian: \(x = r \cos(\theta)\), \(y = r \sin(\theta)\)
• From Cartesian to polar: \(r = \sqrt{x^2 + y^2}\), \(\theta = \arctan(y/x)\) (adjusted based on quadrant)

Polar Curves

A polar curve is a graphical representation of equations expressed in terms of polar coordinates. These curves are described by equations that relate the radial distance \(r\) to the angle \(\theta\). They often produce shapes and patterns that are not easily represented with Cartesian equations.

Consider the circle equations given in the exercise:

• \(r = 3 \cos \theta\): This describes a circle centered at \((3/2, 0)\) with a radius \(3/2\). The classical symmetry of this circle around the horizontal axis makes it straightforward to analyze using polar coordinates.
• \(r = \sin \theta\): Represents a circle centered at \((0, 1/2)\) with radius \(1/2\). Its vertical offset is easily expressed in polar form.

Polar curves can be particularly insightful for visualizing and describing bounded or rotational regions. They are essential for solving problems within circular boundaries and making use of symmetry properties that simplify calculus operations.

Inequalities in Trigonometry

Inequalities in trigonometry involve conditions or restrictions on trigonometric functions. These are crucial in defining areas of interest within given curves or regions, especially in polar coordinates.

In our exercise, the inequalities are used to determine a region between two polar curves. Specifically, one that is inside the circle given by \(r = 3 \cos \theta\) but outside the circle \(r = \sin \theta\). The inequalities help outline the specific section of the plane where these conditions hold true:

• \(\sin \theta \leq r\) ensures that the point \((r, \theta)\) is outside or on the boundary of the smaller circle.
• \(r \leq 3 \cos \theta\) ensures that the point is inside or on the boundary of the larger circle.

Combining these two inequalities lets us use set-builder notation to accurately describe the desired region. Inequalities in trigonometry are vital in problem-solving, allowing us to delineate regions within a plane by understanding and applying trigonometric relationships.