Problem 32 - Essential Calculus

Question

Sketch the strophoid \(r=\sec \theta-2 \cos \theta,\) \(-\frac{\pi}{2}<\theta<\frac{\pi}{2}\). Convert this equation to rectangular coordinates. Find the area enclosed by the loop.

Step-by-Step Solution

Answer

The rectangular equation is \(x=y\tan(\theta)-2y\). The area enclosed by a loop can be calculated by determining the bounds for \(\theta\) from the plot and substituting them into the equation for the area in polar coordinates.

Step 1: Sketching the Polar Curve

Plot the polar function \(r=\sec \theta-2 \cos \theta,\) for \(-\frac{\pi}{2}<\theta<\frac{\pi}{2}\). This can be done by making a table and calculating \(r\) for different \(\theta\) values and then plotting them on polar graph paper.

Step 2: Convert to Rectangular Coordinates

Using the transformation \(r = x\cos(\theta) = y\sin(\theta)\), the equation becomes \(x = y\tan(\theta)- 2y\). This can be done by plugging in the identity \(sec(\theta) = 1/cos(\theta)\) and rearranging the equation.

Step 3: Calculate the area of the Loop

Once we have the plot, we can determine the bounds for \(\theta\) which form a loop and Integrate using the formula for the area in polar coordinates \(A = 1/2 \int_{a}^{b} r^2 d\theta\), where \(a\) and \(b\) are the bounds determined from the plot.

Key Concepts

Rectangular Coordinates

When we talk about rectangular coordinates, we are dealing with a traditional system that uses a grid to pinpoint locations. In this system, each point is identified after two perpendicular lines: the x-axis (running horizontally) and the y-axis (running vertically). Each point has a value of

• an x-coordinate (showing the horizontal distance from the origin),
• and a y-coordinate (showing the vertical distance from the origin).

This creates a coordinate pair \((x, y)\). It's like saying where you are in a city gridmap with street names. Rectangular coordinates are essential for graphing equations and understanding geometric relationships on a flat surface.For example, when you convert an equation from polar to rectangular form, you're translating it onto this flat surface using the relations:

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

These identities help us move from the circular nature of polar coordinates to the grid-like nature of rectangular coordinates.

Strophoid

A strophoid is a type of polar curve that has a characteristic loop shape or double loop, forming symmetrical patterns. It's an interesting curve because it combines elements of geometry and algebra in its formation.The equation for a strophoid, such as \(r = \sec\theta - 2\cos\theta\), describes a shape that generates a clear loop when illustrated. Here's what happens:

• The function takes "secant" and "cosine" into account to express radial distance \(r\) from the pole.
• Its loop occurs within a certain range of \(\theta\), i.e., from \(-\frac{\pi}{2}<\theta<\frac{\pi}{2}\).

By sketching these points on a polar graph, you can visualize the strophoid in its entirety. The points within this range cause the curve to make its dramatic loop.Strophoids are fascinating in mathematics because they provide examples of curves that are not always intuitive but have symmetrical and repetitive nature, showcasing the beauty of mathematical formations.

Polar to Rectangular Conversion

To convert from polar coordinates to rectangular coordinates, you use specific trigonometric identities. Polar coordinates are defined by two values: the radius \(r\) (how far out you go) and the angle \(\theta\) (which direction). On the other hand, rectangular coordinates represent points on the plane through an \(x\) and \(y\) pair.The conversion process relies on these identities:

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

Let's consider converting the polar equation of a strophoid. Using the identity \(\sec(\theta) = \frac{1}{\cos(\theta)}\), and given \(r = \sec \theta - 2 \cos \theta\), we can systematically substitute and rearrange to find expressions in terms of \(x\) and \(y\).Even though the conversion might appear cumbersome, it offers insights into the relationship between different systems of coordinate descriptions. It's a fundamental mathematical skill, especially useful in fields such as physics and engineering, where changing coordinate systems can simplify problem-solving significantly.