Problem 36 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 33-38, sketch the given curves and find their points of intersection. \(r=5, r=\frac{5}{1-2 \cos \theta}\)

Step-by-Step Solution

Answer

The curves intersect at \((5, \frac{\pi}{2})\) and \((5, \frac{3\pi}{2})\).

Step 1: Understand the Given Polar Equations

We have two polar equations: \(r = 5\) and \(r = \frac{5}{1-2 \cos \theta}\). The first equation represents a circle with radius 5, and the second equation is a limacon for which we need to find the points of intersection with the circle.

Step 2: Set Equations Equal to Find Intersection Points

Equate the two equations to find the intersection points: \(5 = \frac{5}{1-2 \cos \theta}\). This equation must be solved to find the values of \(\theta\) where the curves intersect.

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

To solve \(5 = \frac{5}{1-2 \cos \theta}\), first multiply both sides by \(1 - 2 \cos \theta\) to eliminate the fraction: \(5(1 - 2 \cos \theta) = 5\). Simplify to find \(1 - 2\cos \theta = 1\), which leads to \(-2 \cos \theta = 0\) and \(\cos \theta = 0\).

Step 4: Determine Values of \(\theta\)

Since \(\cos \theta = 0\) for \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\), these are the angles at which intersection occurs.

Step 5: Substitute \(\theta\) to Find Corresponding \(r\)

For both angles \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\), substitute back into \(r = 5\) to get \(r = 5\). Therefore, the points are \((5, \frac{\pi}{2})\) and \((5, \frac{3\pi}{2})\).

Step 6: Plot and Confirm the Intersections

Sketch both the circle with radius 5 and the limacon given by \(r = \frac{5}{1-2 \cos \theta}\) to visually confirm the intersection points \((5, \frac{\pi}{2})\) and \((5, \frac{3\pi}{2})\).

Key Concepts

Intersection Points

Intersection points are the coordinates where two or more curves meet or cross each other. In polar coordinates, these points are given as \((r, \theta)\), where \(r\) is the radius, and \(\theta\) is the angle in radians measured from the polar axis. Finding intersection points can involve:

• Setting the equations of the curves equal to one another
• Solving for the variable \(\theta\)
• Substituting back to find the corresponding \(r\)

In our exercise, we equated \(r = 5\) and \(r = \frac{5}{1-2 \cos \theta}\) to determine where these curves intersect. Solving the equation \(5 = \frac{5}{1-2 \cos \theta}\) leads to \(\cos \theta = 0\). This helps us find the angles \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\), which match up with radius \(r = 5\). This gives intersection points \((5, \frac{\pi}{2})\) and \((5, \frac{3\pi}{2})\).

Polar Equations

Polar equations are a way of expressing mathematical relationships in polar coordinates. They are different from Cartesian equations in that they use the radius \(r\) and angle \(\theta\) to describe points in a plane. Understanding polar equations can help in visualizing complex curves such as spirals, circles, and limaçons. To interpret a polar equation like \(r = \frac{5}{1-2 \cos \theta}\), it's essential to analyze the function on the right side of the equation. In general:

• If the equation is a simple constant like \(r = 5\), it depicts a circle of radius 5.
• If the equation involves trigonometric functions, it often represents more complex shapes such as a limaçon.

These equations play a crucial role in the intersection point problems by providing the conditions under which curves intersect in a polar plane.

Limacon

A limacon is a type of polar curve shaped similar to a snail shell (depending on the specific parameter values). It gets its distinctive shape from its equation, usually of the form \(r = a + b \cos \theta\) or \(r = a + b \sin \theta\). In our case, the equation \(r = \frac{5}{1-2 \cos \theta}\) represents a limacon. Key features of a limacon include:

• If \(a = b\), the limacon passes through the pole (origin) and forms a cardioid.
• If \(|a| < |b|\), it has an inner loop.
• If \(|a| > |b|\), it forms a dimpled or convex shape without a loop.

These features influence how the curve may intersect with other curves, such as the circle described by \(r = 5\). Understanding these shapes is critical when solving for their intersection points.

Trigonometric Functions

Trigonometric functions, like sine and cosine, are fundamental when working with polar equations because they determine the angular positioning of points on these curves. They relate the angle \(\theta\) to various transformations of the radius \(r\). In the task at hand, the use of \(\cos \theta\) in the limacon equation \(r = \frac{5}{1-2 \cos \theta}\) helps to describe how the radius changes with the angle. Specific angles, where cosine achieves certain values, simplify solving the equation:

• \(\cos \theta = 0\) occurs at angles \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\), crucial for finding the intersection points.
• Properties like symmetry and periodicity of trigonometric functions affect the shape and the position of polor curves.

Mastering these functions enables you to manipulate polar equations effectively and understand their geometric implications on a polar grid.