Problem 20 - Calculus

Question

Find the points where the two curves meet. \(r=1+\cos \theta\) and \(r=1-\cos \theta\)

Step-by-Step Solution

Answer

The curves meet at points \((1, \frac{\pi}{2})\) and \((1, \frac{3\pi}{2})\).

Step 1: Understanding the Problem

We are given two polar coordinate equations: \(r=1+\cos \theta\) and \(r=1-\cos \theta\). Our task is to find the points where these two curves meet, i.e., where they have the same \(r\) value for the same \(\theta\) value.

Step 2: Setting Equations Equal

To find the points of intersection, set the two equations equal to each other: \(1+\cos \theta = 1-\cos \theta\).

Step 3: Solving for \(\cos \theta\)

Simplify the equation \(1+\cos \theta = 1-\cos \theta\) by subtracting 1 from both sides: \(\cos \theta = -\cos \theta\).

Step 4: Isolating \(\cos \theta\)

Add \(\cos \theta\) to both sides to isolate \(\cos \theta\): \(2\cos \theta = 0\).

Step 5: Finding Angle \(\theta\)

Divide both sides by 2 to solve for \(\cos \theta\): \(\cos \theta = 0\). Since \(\cos \theta = 0\), \(\theta\) can be \(\theta = \frac{\pi}{2}\) or \(\theta = \frac{3\pi}{2}\).

Step 6: Calculating \(r\) for \(\theta = \frac{\pi}{2}\)

Substitute \(\theta = \frac{\pi}{2}\) into either original equation to find \(r\). Using \(r = 1 + \cos(\theta)\), we get \(r = 1 + \cos(\frac{\pi}{2}) = 1\).

Step 7: Checking \(r\) for \(\theta = \frac{3\pi}{2}\)

Substitute \(\theta = \frac{3\pi}{2}\) into either original equation to find \(r\). Using \(r = 1 + \cos(\theta)\), we get \(r = 1 + \cos(\frac{3\pi}{2}) = 1\).

Step 8: Conclusion

Both \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\) yield \(r = 1\). Therefore, the curves intersect at the points \((1, \frac{\pi}{2})\) and \((1, \frac{3\pi}{2})\).

Key Concepts

Polar Coordinates

Polar coordinates provide a powerful way to describe locations in the plane, especially for problems involving circles or angles. Unlike Cartesian coordinates which use \(x\) and \(y\), polar coordinates describe a point with a radius \(r\) and an angle \(\theta\).

• **Radius (r):** This is the distance from the origin to the point. It can be thought of as how far out you go from the center.
• **Angle (\theta):** This is the angle measured from the positive x-axis, usually moving counter-clockwise.

For the given problem, the polar equations \(r = 1 + \cos \theta\) and \(r = 1 - \cos \theta\) describe curves that are effectively circles displaced along the x-axis. Polar coordinates shine here because they can simplify the representation of curves that would otherwise yield more complicated Cartesian equations.

Trigonometric Equations

Solving trigonometric equations often involves manipulating expressions to isolate trigonometric functions such as \(\cos, \sin\), or \(\tan\). In this problem, we solve equations involving \(\cos \theta\). When we set \(r = 1 + \cos \theta\) equal to \(r = 1 - \cos \theta\), we derive \(\cos \theta = -\cos \theta\), a situation calling for clear problem solving.

• This equation simplifies to \(2\cos \theta = 0\) leading directly to \(\cos \theta = 0\).
• The solution for \(\theta\), where \(\cos \theta = 0\), occurs when \(\theta = \frac{\pi}{2}\) or \(\theta = \frac{3\pi}{2}\).

Understanding these properties and how they translate from algebraic manipulations to geometric interpretations is essential for accurately finding intersections between curves.

Problem-Solving Steps

Breaking down a mathematical problem into manageable steps is crucial for clarity. For the task of finding where two polar curves intersect, the approach involves equating and solving. Here’s how you can logically follow through:

• **Understand the problem:** Recognize the curves described by their polar equations. We are required to find common points \(\theta\) where both give the same \(r\).
• **Set and simplify equations:** Start by equating the given functions, leading you to solve for specific values of \(\theta\).
• **Solve trigonometric identities:** Solve \(\cos \theta = -\cos \theta\) by isolating terms and using known trigonometric angles.
• **Find corresponding radius values:** Input these angles back into one of the original equations to find \(r\).

By following these steps, you effectively tackle curve intersection problems using polar coordinates, ensuring each step logically progresses from the last.