Problem 75 - A Graphical Approach to Precalculus with Limits

Question

Find the polar coordinates of the points of intersection of the given curves for the specified interval of \(\theta\). $$r=2+\sin \theta, r=2+\cos \theta ; 0 \leq \theta<2 \pi$$

Step-by-Step Solution

Answer

The polar coordinates are \((2.707, \frac{\pi}{4})\) and \((1.293, \frac{5\pi}{4})\).

Step 1: Equate the Two Polar Equations

We are given two polar equations: \( r = 2 + \sin \theta \) and \( r = 2 + \cos \theta \). To find their points of intersection, we set the two expressions for \( r \) equal to each other: \( 2 + \sin \theta = 2 + \cos \theta \). Simplifying, this gives \( \sin \theta = \cos \theta \).

Step 2: Solve the Trigonometric Equation

The equation \( \sin \theta = \cos \theta \) can be rewritten using the identity \( \tan \theta = 1 \), leading to \( \theta = \frac{\pi}{4} + n\pi \), where \( n \) is an integer. In the interval \( 0 \leq \theta < 2\pi \), the solutions are \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{5\pi}{4} \).

Step 3: Substitute \(\theta\) into One of the Polar Equations

For each \( \theta \) value found, substitute into either \( r = 2 + \sin \theta \) or \( r = 2 + \cos \theta \) to find the corresponding \( r \). Start with \( \theta = \frac{\pi}{4} \): \( r = 2 + \sin\frac{\pi}{4} = 2 + \frac{\sqrt{2}}{2} = 2.707 \). Now for \( \theta = \frac{5\pi}{4} \): \( r = 2 + \sin\frac{5\pi}{4} = 2 - \frac{\sqrt{2}}{2} = 1.293 \).

Step 4: Write the Polar Coordinates

The polar coordinates for the points of intersection are \( \left(2.707, \frac{\pi}{4}\right) \) and \( \left(1.293, \frac{5\pi}{4}\right) \).

Key Concepts

Points of Intersection

When dealing with polar coordinates, finding the points of intersection between curves involves setting two given equations equal to each other. This is similar to finding intersections in Cartesian coordinates, but requires consideration of the polar system.
In our problem, we have two polar equations: \( r = 2 + \sin \theta \) and \( r = 2 + \cos \theta \). By equating them, we find the condition for intersection: \( \sin \theta = \cos \theta \). This sets up the opportunity to solve for the specific values of \( \theta \) where these curves meet. The intersection points are places where both curves have the same radius \( r \) at a particular \( \theta \), providing the polar coordinates for these intersections.
Understanding these intersections in polar coordinates involves translating the idea of overlapping lines or curves into the language of angles and radii. This is crucial because it allows us to visualize how two polar curves overlap within their specified range of \( \theta \).

Trigonometric Equations

Trigonometric equations often arise when equating polar equations because they involve functions like sine and cosine. The given example, \( \sin \theta = \cos \theta \), requires transforming it into an equivalent trigonometric form that is easier to solve.
By recognizing that for angles where \( \sin \theta = \cos \theta \), tangent can be employed; thus, \( \tan \theta = 1 \). This leads us to solutions of the form \( \theta = \frac{\pi}{4} + n\pi \), where \( n \) is an integer.
In a typical exercise, we determine which solutions fit within the given range, such as \( 0 \leq \theta < 2\pi \). Evaluating this within the interval helps us focus on the manageable specific answers, which in turn apply directly to any problem-solving requirements.
Learning to convert between trigonometric forms and solve these equations efficiently is key to handling polar problems successfully.

Polar Equations

Polar equations describe curves using a system based on angles and distances from a central point, or pole. Unlike Cartesian coordinates, which use horizontal and vertical distances, polar equations give positions based on radii and angles.
In polar coordinates, an equation like \( r = 2 + \sin \theta \) defines a specific curve by describing how the radius \( r \) changes as the angle \( \theta \) varies. This creates different types of spirals, circles, or rose shapes, depending on the equations involved. Our example involves two such curves: \( r = 2 + \sin \theta \) and \( r = 2 + \cos \theta \), each forming a different but overlapping pattern.
Comparing these patterns helps identify intersections and understand the graphing behavior of polar equations. Unlike Cartesian settings, where straight lines or parabolas are common, polar graphs can be more visually intricate and fascinating.
Learning how these equations work enhances mathematical insight, showcasing a different spectrum of geometry that gives rise to symmetrical and rotational arrangements typical in polar graphing.