Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of the given cylindrical or spherical equation. $$ r=2 \sin 2 \theta $$

Step-by-Step Solution

Answer

The graph is a four-petal rose centered at the origin.

Step 1: Identify the type of equation

The equation given is in polar form, specifically in terms of the radial coordinate \( r \) as a function of the angular coordinate \( \theta \). The equation is \( r = 2 \sin 2\theta \). Our task is to sketch the graph based on this polar equation.

Step 2: Analyze the Equation Components

The given equation \( r = 2 \sin 2\theta \) involves the sine function, which is periodic. Notice that the argument of the sine function is \( 2\theta \), which means there will be multiple cycles happening as \( \theta \) goes from \( 0 \) to \( 2\pi \). The amplitude of the sine is 2, meaning \( r \) will range from 0 to 2.

Step 3: Determine Key Points

To plot key points, consider the values of \( \theta \) that simplify the sine function, such as \( \theta = 0, \frac{\pi}{4}, \frac{\pi}{2}, \ldots \). Calculate \( r \) for some of these angles:- \( \theta = 0, r = 2\sin(0) = 0 \) - \( \theta = \frac{\pi}{4}, r = 2\sin(\frac{\pi}{2}) = 2\)- \( \theta = \frac{\pi}{2}, r = 2\sin(\pi) = 0\)- \( \theta = \frac{3\pi}{4}, r = 2\sin(\frac{3\pi}{2}) = -2\) (interpreted in opposite direction)- Repeat in the interval \( \pi \leq \theta \leq 2\pi \).

Step 4: Sketch One Cycle of the Graph

Plot the points calculated in Step 3 and remember to connect them smoothly because the graph is a continuous polar curve. Each full sine wave in \( r = 2\sin 2\theta \) occurs over a \( \pi \) interval.

Step 5: Complete Graphing Within One Rotation

The entire graph will complete within \( \theta = 0 \) to \( 2\pi \), completing two full cycles of the sine function because of the \( 2\theta \) argument. The pattern will mimic a rose curve with four petals due to the symmetry and periodicity of \( 2\sin 2\theta \).

Step 6: Verify the Sketch Completeness

Ensure that each 'petal' of the rose is plotted symmetrically around the origin with correct radius. Each cycle produces two petals; hence \( r = 2 \) at \( \theta = \frac{\pi}{4} \) and \( \frac{5\pi}{4} \), and \( r = -2 \) at \( \theta = \frac{3\pi}{4} \) and \( \frac{7\pi}{4} \).

Key Concepts

Graphing Polar Equations

Polar equations are different from the Cartesian equations because they involve coordinates based on angles and distances from a point called the pole. In polar coordinates, every point on a plane is determined by a distance and an angle from a fixed point. Polar equations use the notation \( (r, \theta) \), where \( r \) is the radius and \( \theta \) is the angle measured from the positive x-axis.To graph polar equations, like \( r = 2 \sin 2\theta \), begin by understanding how \( r \) changes as \( \theta \) varies from 0 to \( 2\pi \). Each value of \( \theta \) gives a corresponding \( r \), which describes a point on the polar plot. Analyzing key angles helps balance accuracy with ease of sketching. The graph often has a periodic pattern, produced by evaluating the function over one or more full cycles.

• Identify the type of graph (e.g., circle, rose curve)
• Select key angles that simplify computation
• Plot symmetrical points for smooth curves

Periodic Functions

Periodic functions are mathematical functions that repeat their values in regular intervals or periods. A familiar example is the sine function, which repeats every \( 2\pi \) radians. Understanding periodicity is crucial for graphing and analyzing polar equations like \( r = 2 \sin 2\theta \), where the periodic nature creates repeating patterns.Characteristics of periodic functions include:

• Repetition after a fixed interval \( T \)
• The smallest positive \( T \) is called the fundamental period
• In polar graphs, the periodicity leads to symmetrical shapes

For \( r = 2 \sin 2\theta \), the function \( \sin 2\theta \) implies a period of \( \pi \), meaning every \( \pi \) radians repeat the behavior of \( r \). This periodic behavior results in multiple cycles and a pattern on the polar graph.

Sine Function

The sine function is a basic trigonometric function that shows the y-coordinate of a point on the unit circle as you move counterclockwise from the positive x-axis. Sine is an essential function in polar equations, as it describes the wave-like pattern of many graphs. For \( r = 2 \sin 2\theta \), it's valuable to explore the sine function's behavior.Key properties:

• Sine waves alternate between -1 and 1
• Points of interest often occur at \( 0, \frac{\pi}{2}, \pi, ... \)
• Amplitude affects the maximum and minimum value of \( r \)

In the case of \( r = 2 \sin 2\theta \), the sine function has been scaled by 2, meaning \( r \) will reach a maximum of 2 and a minimum of -2. The argument \( 2\theta \) adjusts the frequency of peaks and troughs, crucial for determining the graph's exact shape and symmetry.

Rose Curve

A rose curve is a delightful and aesthetically pleasing graph that emerges from polar equations of specific forms. Polar equations in the form \( r = a \sin n\theta \) or \( r = a \cos n\theta \) often result in a rose curve, with "petals" that appear due to the periodic nature and symmetry of these functions.Understanding rose curves involves:

• Identifying the number of petals: If \( n \) is even, there are \( 2n \) petals, if odd, \( n \) petals
• Symmetry: Rose curves are symmetrical around the pole
• Amplitude sets the length of petals

For the equation \( r = 2 \sin 2\theta \), we expect a rose pattern with four petals. This is because the double angle \( 2\theta \) creates two full cycles (hence twice the number of apparent cycles), and the sine function forms symmetrical lobes or petals as \( \theta \) varies from \( 0 \) to \( 2\pi \). Each petal is structured around key angles you identified, offering elegant and visually symmetric polar plots.