Problem 97 - Precalculus

Question

In calculus, when we need to find the area enclosed by two polar curves, the first step consists of finding the points where the curves coincide. Find the points of intersection of the given curves. $$r=1-\sin \theta \quad \text { and } \quad r=1+\cos \theta$$

Step-by-Step Solution

Answer

Intersection points: \((1 + \frac{\sqrt{2}}{2}, \frac{3\pi}{4})\) and \((1 - \frac{\sqrt{2}}{2}, \frac{7\pi}{4})\).

Step 1: Understand the Problem

We need to find the points where two polar equations intersect. The given equations are \( r = 1 - \sin \theta \) and \( r = 1 + \cos \theta \).

Step 2: Set Equations Equal

To find the points of intersection, set the equations for \( r \) equal to each other:\[ 1 - \sin \theta = 1 + \cos \theta \]

Step 3: Simplify the Equation

Subtract 1 from both sides to simplify the equation:\[ -\sin \theta = \cos \theta \]

Step 4: Rearrange and Solve

Divide both sides by \( \cos \theta \) to isolate \( \tan \theta \):\[ \tan \theta = -1 \]

Step 5: Solve for \( \theta \)

The solution for \( \tan \theta = -1 \) is where \( \theta = \frac{3\pi}{4} \) or \( \theta = \frac{7\pi}{4} \) within \( [0, 2\pi) \).

Step 6: Find Corresponding \( r \) Values

Substitute the \( \theta \) values back into either polar equation to find \( r \). Using \( r = 1 - \sin \theta \):For \( \theta = \frac{3\pi}{4} \), \( r = 1 - (-\frac{\sqrt{2}}{2}) = 1 + \frac{\sqrt{2}}{2} \).For \( \theta = \frac{7\pi}{4} \), \( r = 1 - (\frac{\sqrt{2}}{2}) = 1 - \frac{\sqrt{2}}{2} \).

Key Concepts

Points of Intersection

In polar coordinates, finding points of intersection is a crucial step when determining where two polar curves meet. These are the points where the values of \( r \) are equal for both curves at the same angle \( \theta \). For example, consider the polar equations \( r = 1 - \sin \theta \) and \( r = 1 + \cos \theta \). To find where these curves intersect, set the equations equal:
\( 1 - \sin \theta = 1 + \cos \theta \).
This equation helps us identify where the two curves share the same distance from the origin for specific angles.

By solving this simplified equation, such as through algebraic manipulation or trigonometric identities, we can determine which angles \( \theta \) result in the same \( r \) value for both curves. In our example, we find \( \theta = \frac{3\pi}{4} \) and \( \theta = \frac{7\pi}{4} \), angles where both functions intersect.

Trigonometric Equations

Trigonometric equations play a significant role in solving polar coordinate problems, particularly when finding intersections. A trigonometric equation, like \( \tan \theta = -1 \), involves a trigonometric function set to a specific value. Solving these equations often provides the angle(s) \( \theta \) where certain conditions are met.

• To solve \( \tan \theta = -1 \), identify angles where the tangent function equals \(-1\).
• Use the unit circle or function properties to find these angles.

In the example, we found the solutions \( \theta = \frac{3\pi}{4} \) and \( \theta = \frac{7\pi}{4} \), which are specific angles within the interval \([0, 2\pi)\) where the tangent function fulfills this equation. This process is fundamental when dealing with polar graphs, as these angles determine where intersection occurs on the curves.

Graphing Polar Coordinates

Graphing polar coordinates involves plotting points found using polar equations, which express the radius \( r \) in terms of the angle \( \theta \). Each point is given in the form \((r, \theta)\), where \( r \) is the distance from the origin, and \( \theta \) is the angle from the positive x-axis.

Visualizing polar curves can be challenging, but it's crucial, especially when analyzing intersections:

• Begin by sketching individual curves separately.
• For the example curves, sketch where \( r = 1 - \sin \theta \) and \( r = 1 + \cos \theta \) individually cover the polar plane.
• Highlight the points corresponding to \( \theta = \frac{3\pi}{4} \) and \( \theta = \frac{7\pi}{4} \), as these indicate intersections.

By graphing, you can better understand the relationship between curves, how they intersect, and the areas they encompass, making it easier to predict and analyze solutions visually.