Problem 32 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

The curves intersect at \((1, 0)\) and \((1, \pi)\) in polar coordinates.

Step 1: Understand the Polar Equations

We have two polar equations: \( r = 1 + \sin \theta \) and \( r = 1 - \sin \theta \). Our goal is to find the intersection points, meaning the values of \( \theta \) for which both equations give the same \( r \) value. Thus, set \( 1 + \sin \theta = 1 - \sin \theta \).

Step 2: Solve the Equations for \( \sin \theta \)

Set the equations equal: \( 1 + \sin \theta = 1 - \sin \theta \). By simplifying, subtract \( 1 \) from both sides to get \( \sin \theta = -\sin \theta \).

Step 3: Analyze \( \sin \theta = -\sin \theta \)

This equation implies \( 2\sin \theta = 0 \) when bringing \( \sin \theta \) to one side, yielding \( \sin \theta = 0 \). Determine the \( \theta \) values where \( \sin \theta = 0 \), specifically at \( \theta = 0 \), and \( \theta = \pi \), within the range \( [0, 2\pi) \).

Step 4: Check \( r \) Values for Intersections

Substitute \( \theta = 0 \) and \( \theta = \pi \) into the original equations to find \( r \) values. For \( \theta = 0 \), \( r = 1 + \sin 0 = 1 \) and \( r = 1 - \sin 0 = 1 \). For \( \theta = \pi \), \( r = 1 + \sin \pi = 1 \) and \( r = 1 - \sin \pi = 1 \).

Step 5: Identify Intersection Points

Thus, the points of intersection are at \((1, 0)\) and \((1, \pi)\) in polar coordinates. Each \( \theta \) value provided the same \( r \) value for both equations, confirming the intersections.

Key Concepts

Understanding Intersection of Curves

When dealing with polar coordinates, the intersection of curves occurs where two conditions are met: both curves have the same radial distance, known as the radius \(r\), from the pole (origin) at the same angle \(\theta\). In simpler terms, the intersection is where both equations equal the same \(r\) for certain values of \(\theta\).To find these points, you can equate the two polar equations and solve for \(\theta\). This gives the angles where the curves will intersect. However, it's crucial to remember that such intersections must be verified by checking the radial values from both equations. Polar coordinates might show the same point differently (like negative \(r\)), so always cross-verify the \(r\) values through substitution in the original equations. Here, the equations are set as:

• \(r=1+\sin \theta\)
• \(r=1-\sin \theta\)

Solving these provides the intersections, effectively the heart of this exercise.

Exploring Polar Equations

Polar equations are a mathematical way to express a curve on a plane using the polar coordinate system. In this system, each point on the plane is determined by a distance \(r\) from a central point (called the pole), and an angle \(\theta\) from a fixed direction, usually the positive x-axis.Unlike the Cartesian system where points are represented as \((x, y)\), in polar coordinates they are \((r, \theta)\). This system is especially useful for curves that are circular or have some kind of rotational symmetry. The equations here use trigonometric functions (sine and cosine) to dictate how \(r\) varies with \(\theta\). For the given problem:

• The equation \(r = 1 + \sin\theta\) suggests the curve fluctuates above and below 1 due to the sine function's variation.
• For \(r = 1 - \sin\theta\), the sine component decreases \(r\) in a complementary fashion.

Understanding these properties helps visualize the curves and identify possible intersections.

Utilizing Trigonometric Identities

Trigonometric identities play a crucial role in solving and simplifying equations involving sine, cosine, and other trigonometric functions. These identities are fundamental rules that relate different trigonometric functions to each other, and they can greatly simplify the task of finding intersections of polar curves.One of the key identities used in the exercise is recognizing that \(\sin\theta = -\sin\theta\) implies \(\sin\theta = 0\). This identity leads us to conclude that the values for \(\theta\) where this holds true are multiples of \(\pi\), specifically \(\theta = 0\) and \(\theta = \pi\) within the range \([0, 2\pi)\). This is because the sine function equals zero at these angles, which can be verified by substituting back into the polar equations to ensure both give the same \(r\).By mastering these identities, you can reveal solutions that might not be immediately apparent, especially in problems involving polar equations.