Problem 32 - Thomas Calculus

Question

The nephroid of Freeth Graph the nephroid of Freeth: $$r=1+2 \sin \frac{\theta}{2}.$$

Step-by-Step Solution

Answer

Plot the polar curve for \( r = 1 + 2\sin \frac{\theta}{2} \) using \( \theta \) from 0 to \( 2\pi \).

Step 1: Understanding the Equation

The given equation is in polar coordinates where \( r \) is a function of \( \theta \). The nephroid of Freeth is a type of curve created by the rolling of a circle. In this case, we will plot \( r = 1 + 2 \sin \frac{\theta}{2} \) for \( \theta \) ranging from \( 0 \) to \( 2\pi \).

Step 2: Determine Characteristics of the Function

Notice how the term \( \sin \frac{\theta}{2} \) affects the radius. As \( \theta \) increases, the value of \( r \) oscillates between 1 and 3 because \( \sin \theta \) varies between -1 and 1. This creates the unique kidney-shaped curve of the nephroid.

Step 3: Calculate Key Points

To sketch the graph, calculate key points such as at \( \theta = 0, \theta=\pi \), and \( \theta = 2\pi \). For \( \theta = 0 \), \( r = 1 + 2 \cdot 0 = 1 \). For \( \theta = \pi \), \( r = 1 + 2 \cdot 1 = 3 \). For \( \theta = 2\pi \), we return to \( r = 1 \).

Step 4: Plot the Curve

Using the data points and understanding that the curve is smooth and periodic over \( 0 \leq \theta < 2\pi \), draw the nephroid using these values. The shape is symmetric about the x-axis.

Step 5: Graphical Representation

Use graphing software or graph paper to plot \( r = 1 + 2\sin \frac{\theta}{2} \) over the full range of \( \theta \). The result is a kidney-shaped curve that touches the x-axis at its widest points.

Key Concepts

Polar Coordinates

Polar coordinates offer an intriguing way to describe the position of a point in the plane. Instead of using

• rectangular coordinates
• \((x, y)\), which are derived from right-angle intersections of axes,
• polar coordinates specify a point through a radius and an angle.

In this system, a point is described by \((r, \theta)\), where \(r\) is the distance from the origin and \(\theta\) is the angle from the positive x-axis.

This is particularly useful in cases where symmetry or periodicity is involved, like in the case of the nephroid of Freeth. The radius \(r\) is not constant but changes with \(\theta\). Understanding this relationship is key to graphing polar equations.

Graphing Polar Equations

Polar equations, like the nephroid of Freeth represented by \(r = 1 + 2 \sin \frac{\theta}{2}\), provide a different approach to plotting curves. Unlike Cartesian graphs where you consider \(x\) and \(y\),

• polar equations focus on \(r\) and \(\theta\).
• To graph such an equation, you calculate the value of \(r\) over a range of angles.

This involves determining the characteristics of the equation, observing how changes in \(\theta\) affect \(r\), and plotting these points in succession.

For \(r = 1 + 2 \sin \frac{\theta}{2}\), as \(\theta\) varies from \(0\) to \(2\pi\), \(r\) oscillates, allowing the curve to reform symmetrically. By plotting key angles like \(\theta = 0, \theta = \pi\), and \(\theta = 2\pi\), and considering the continuous nature of \(r\), you can sketch the kidney-shaped nephroid.

Trigonometric Functions

Trigonometric functions, particularly sine and cosine, form the backbone of many polar equations, including the nephroid of Freeth. They describe primary relationships based on angles and are periodic functions.

In the equation \(r = 1 + 2 \sin \frac{\theta}{2}\),

• \(\sin \frac{\theta}{2}\) can take values between -1 to 1, influencing the range of \(r\) as \(\theta\) changes,
• creating oscillations that mirror the periodic nature of \(\sin \theta\).
• Through adjustments in amplitude and phase (like multiplying or shifting),
• trigonometric functions help yield the specific shape of polar plots, reflecting symmetry and periodicity.

Understanding these characteristics is vital, as they determine the nature of the curves we graph in polar coordinates.