Question

Graph the polar function on the given interval. \(r=\sin (3 \theta), \quad[0, \pi]\)

Short answer

The polar graph of \( r = \sin(3\theta) \) over \([0, \pi]\) has three loops.

Step-by-step solution

Understand Polar Coordinates

Polar coordinates represent points in the plane using a distance from the origin and an angle from the positive x-axis. For the given function \( r = \sin(3\theta) \), \( r \) is the radial coordinate, and \( \theta \) is the angular coordinate.

Analyze the Function

Observe that \( r = \sin(3\theta) \) is a sinusoidal function in terms of \( \theta \). Multiplying \( \theta \) by 3 indicates that the period of the sine function will change, resulting in more loops within a given interval.

Determine Key Points

Identify key points where \( r = 0 \), \( r = 1 \), and \( r = -1 \) within the interval \([0, \pi]\). Use these points to help sketch the graph:- \( r = 0 \) when \( \theta = 0, \frac{\pi}{3}, \frac{2\pi}{3}, \pi \).- \( r = 1 \) when \( 3\theta = \frac{\pi}{2} + 2k\pi \rightarrow \theta = \frac{\pi}{6}, \frac{5\pi}{6} \).- \( r = -1 \) when \( 3\theta = \frac{3\pi}{2} + 2k\pi \rightarrow \theta = \frac{\pi}{2} \).

Understand Symmetry and Loops

The function \( r = \sin(3\theta) \) is symmetric about the origin which means it repeats its pattern. There will be three loops embedded within the interval \([0, \pi]\).

Graph the Function

Plot the polar coordinates using the key points identified in Step 3 and the understanding from Step 4. Each loop originates from the origin, extends outwards, and returns to the origin at the intervals calculated.

Verify with a Polar Plot

Use a graphing tool to plot the function \( r = \sin(3\theta) \) within the interval \([0, \pi]\) to ensure accuracy and verify the presence of three symmetrical loops.

Key concepts

Graphing Polar Functions

Graphing polar functions can be both fascinating and distinctive compared to graphing Cartesian functions. In polar coordinates, each point on the plane is defined by a distance from the origin, denoted as \( r \), and an angle \( \theta \) from the positive x-axis. This system is especially useful because it naturally models circular patterns and periodic behaviors.
Consider the polar equation \( r = \sin(3\theta) \). Here, the sine function causes the radius \( r \) to oscillate between -1 and 1 as \( \theta \) varies from 0 to \( \pi \). The factor of 3 indicates that the function will cycle through its pattern more frequently than a typical sine function. In fact, it will complete three full cycles within the interval from 0 to \( \pi \), producing a graph with three distinct loops.
To effectively graph such a function, it is beneficial to determine where the sine function takes important values such as 0, 1, and -1. These key points help outline the graph structure and provide reference angles at which the graph peaks, intersects the origin, or completely turns. By plotting these points on a polar grid, and smoothly connecting them, the complete graph of the polar function can be visualized.

Symmetry in Polar Graphs

Symmetry plays an integral role in understanding and graphing polar equations, as it helps reduce the amount of work needed to plot the entire graph. For the function \( r = \sin(3\theta) \), the symmetry is about the origin, which means that for every point \((r, \theta)\) on the graph, a corresponding point \((-r, \theta + \pi)\) exists.
This characteristic implies that once you understand the behavior of the function in a particular section, you can predict the graph's appearance in other sections without needing additional calculations. This symmetry leads to a predictable pattern, especially helpful for polynomials and trigonometric functions represented in polar form.
Understanding symmetry reduces the complexity involved in plotting or analyzing polar graphs since you can rely on properties already derived for a smaller subset of the function. By identifying symmetrical aspects early, you can more efficiently interpret the image that the polar graph will create.

Sine Function in Polar Equations

The sine function, when utilized in polar equations, introduces oscillating radial patterns within the graph. The basic function \( r = \sin(n\theta) \) results in an interesting and ornate design in polar coordinates. For \( r = \sin(3\theta) \), the number 3 indicates the number of petals the resulting "rose" pattern will have, a classic shape associated with polar graphs.
Sine functions in polar coordinates derive their shape from the nature of the sine wave itself. When multiplied by a constant such as 3 in \( 3\theta \), the function's frequency increases, leading to multiple cycles over the given interval, each displaying one petal of the rose. This pattern occurs due to the way sine functions transition between positive, zero, and negative values as the angle \( \theta \) changes.
The amplitude of the sine function dictates how far from the origin each loop extends, while the number of petals is determined by the coefficient of \( \theta \). This makes sine functions in polar equations a neat example of how algebraic manipulations affect graphical outcomes, showcasing the dynamic mix of shapes that trigonometric functions can create in polar form.