Problem 31 - Thomas Calculus

Question

Find the points of intersection of the pairs of curves in Exercises \(31-38\) . $$ r=1+\cos \theta, \quad r=1-\cos \theta $$

Step-by-Step Solution

Answer

The curves intersect at points \((1, \frac{\pi}{2})\) and \((1, \frac{3\pi}{2})\) in polar coordinates.

Step 1: Convert Polar Equations to Cartesian Coordinates

The given equations are in polar form. The polar equations can be converted to Cartesian coordinates by using the formulas: \ \( x = r\cos \theta \) and \( y = r\sin \theta \). However, it's often easier to handle these in polar form for intersections by manipulating the equations directly. So, here, let's equate them directly in polar form.

Step 2: Equate the Radius Equations

Set the equations equal to each other to find the points where they intersect: \ \( 1 + \cos \theta = 1 - \cos \theta \).

Step 3: Solve the Equation for \( \theta \)

To solve \( 1 + \cos \theta = 1 - \cos \theta \), subtract \( \cos \theta \) from both sides of the equation: \ \( 1 + \cos \theta = 1 - \cos \theta \)\ It becomes: \ \( 2\cos \theta = 0 \). \ Solve to find \( \cos \theta = 0 \). From trigonometric values, \( \theta = \frac{\pi}{2}, \frac{3\pi}{2} \).

Step 4: Substitute Values to Find Intersection Points

Substitute \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \) back into either original equation to find \( r \): \ - If \( \theta = \frac{\pi}{2} \), \( r = 1 - \cos \frac{\pi}{2} = 1 - 0 = 1 \) (also for \( r = 1 + \cos \frac{\pi}{2} = 1 \)). The point is \( (1, \frac{\pi}{2}) \) in polar form.\ - If \( \theta = \frac{3\pi}{2} \), \( r = 1 - \cos \frac{3\pi}{2} = 1 - 0 = 1 \) (also for \( r = 1 + \cos \frac{3\pi}{2} = 1 \)). The point is \( (1, \frac{3\pi}{2}) \) in polar form.

Key Concepts

Polar Coordinates

Polar coordinates are an alternative method to describe the position of a point in a plane using a radius and an angle. Instead of traditional Cartesian coordinates represented by \((x, y)\), polar coordinates are given by \((r, \theta)\), where \(r\) is the distance from the origin and \(\theta\) is the angle formed with the positive x-axis. This system is particularly useful in cases involving circular and spiral paths as it simplifies calculations.

• **Radius \(r\)**: The straight-line distance from the origin to the point.
• **Angle \(\theta\)**: The counterclockwise angle from the positive x-axis to the line segment connecting the origin to the point.

In solving problems involving intersections, polar coordinates help to equate the radius values directly, often making the solution more straightforward than converting everything to Cartesian form.

Graphing Polar Equations

When graphing polar equations, the relation between \(r\) and \(\theta\) defines the shape on the plane. Unlike linear graphs in Cartesian coordinates, graphing in polar form involves plotting based on radius and angle.
The given equations in our example, \(r = 1 + \cos \theta\) and \(r = 1 - \cos \theta\), depict heart-shaped curves known as cardioids.

• For \(r = 1 + \cos \theta\), the graph forms a cardioid with its "heart" facing right.
• For \(r = 1 - \cos \theta\), the shape faces left.

To plot these curves, you can compute several \((r, \theta)\) pairs and translate them into Cartesian coordinates for easier graphing or use graphing technology that handles polar coordinates to visualize them. The intersection points occur where the equations for \(r\) coincide for the same \(\theta\).

Trigonometric Identities

Trigonometric identities play a crucial role in simplifying and solving equations involving trigonometric functions, which are prevalent in polar equations. In the solution step, equating \(1 + \cos \theta = 1 - \cos \theta\) leads to a simplification using the identity related to cosine. Utilizing the identity \(2\cos \theta = 0\) simplifies to \(\cos \theta = 0\).
Key trigonometric identities include:

• **Pythagorean Identities**: \(\sin^2 \theta + \cos^2 \theta = 1\).
• **Even-Odd Identities**: Helps determine angles based on quadrants.

For this example, knowing the cosine values at specific angles like \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\) allowed us to find the intersections effectively, as these angles lead to a zero cosine value.

Cartesian Coordinates

While polar coordinates are highly beneficial for curve intersections, it's often necessary to understand how they relate to the Cartesian system. Cartesian coordinates \((x, y)\) use perpendicular axes to display points, distinct from the circle-like structure of polar.
Understanding the conversion from polar to Cartesian is crucial when switching between graphing methods:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)

This conversion aids in visualizing and verifying the solutions obtained in polar form. For example, from the polar point \((1, \frac{\pi}{2})\), converting to \((x, y)\) yields \((0, 1)\), providing an alternative view of the intersection point on a familiar Cartesian plane. Even though problems can be solved in polar form, checking your results in Cartesian might confirm your solutions' accuracy.