Problem 29 - Precalculus Mathematics for Calculus

Question

Sketch a graph of the polar equation. $$r=\sin 2 \theta$$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Polar Equation

The given polar equation is \( r = \sin 2\theta \). In polar coordinates, \( r \) represents the distance from the origin, and \( \theta \) is the angle. This equation describes how \( r \) changes as \( \theta \) changes.

Step 2: Determine Key Points

We can find key points by evaluating \( \theta \) values. Notice that \( \sin \) has a range between -1 and 1 and it achieves these max/min values at \( 2\theta = \frac{\pi}{2}, \frac{3\pi}{2}, \ldots \). Calculate \( \theta \) for these critical values: \( \theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4} \), giving \( r = 1, -1, 1, -1 \) respectively.

Step 3: Plot Key Points and Symmetry

Plot the points. For \( \theta = \frac{\pi}{4}, \theta = \frac{5\pi}{4} \), \( r = 1 \); for \( \theta = \frac{3\pi}{4}, \theta = \frac{7\pi}{4} \), \( r = -1 \). The graph has symmetry about the origin, creating a four-leaf pattern due to the \( \sin 2\theta \) term.

Step 4: Draw the Complete Graph

Connect the plotted points smoothly, forming a petal-like shape. Each pair of opposite petals occurs every \( \frac{\pi}{2} \) radians with the petals passing through origins point, characterized by intersections when \( r = 0 \) (\( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \)).

Step 5: Check for Cyclic Patterns

Since \( \sin 2\theta \) is periodic with period \( \pi \), the pattern repeats its four-leaf shape fully within the interval \( [0, \pi] \). Ensure that the petals are symmetrical and check that the graph doesn't require additional points beyond one full cycle.

Key Concepts

Graphing Polar Coordinates

Polar coordinates offer a unique way to plot points using a radius and an angle. Unlike rectangular coordinates which use a grid system (x, y), polar coordinates utilize a point's direction and distance from the origin. In our example of the polar equation \( r = \sin 2\theta \), each coordinate \( (r, \theta) \) represents the unique pair of distance and direction from the pole (origin).
When graphing polar coordinates, the steps usually involve:

• Finding critical points by choosing various values of \( \theta \) and calculating the corresponding \( r \).
• Noting that \( r \) can be positive or negative, which determines whether the point lies on the opposite side of the origin.
• Visualizing the curve by connecting these points in a smooth manner.

The graph of \( r = \sin 2\theta \) forms a pattern known as a rose or "four-leaf" clover, with its leaves generated by key points and intermediate \( \theta \). By following these guided steps, you are able to visually represent polar equations easily.

Symmetry in Polar Graphs

Symmetry in polar graphs is an essential feature that helps in simplifying the graphing process. In polar equations like \( r = \sin 2\theta \), symmetry plays a crucial role in understanding and predicting the graph's behavior. There are three potential types of symmetry you can check for:

• **Polar Axis (Horizontal Line through the Origin):** Check by replacing \( \theta \) with \( -\theta \).
• **Line \( \theta = \frac{\pi}{2} \):** Check by replacing \( \theta \) with \( \pi - \theta \).
• **Origin:** Check by replacing \( r \) with \( -r \).

In our function, the equation exhibits symmetry about the origin. This means that for every point \( (r, \theta) \) on the graph, there is a corresponding point \( (-r, \theta + \pi) \), resulting in a pattern that can be anticipated without plotting every single point. Identifying symmetry effectively reduces the amount of work needed when graphing these equations.

Periodicity in Polar Functions

Periodicity is another fundamental characteristic of polar functions, showing how often a pattern repeats as \( \theta \) revolves around the origin. The graph of \( r = \sin 2\theta \) is periodic, repeating every \( \pi \) radians due to the sine component \( \sin 2\theta \).
The pattern observed is that for a complete set of "leaves" in the rose (or petal shape), you need to plot points over the interval \( [0, \pi] \).

• Each complete pattern is known as a cycle.
• Within each cycle, the polar graph will return to its starting state.
• The frequency of repeated patterns informs on the function's periodicity. For this graph, the appearance of the four leaves fully captures one complete cycle of periodicity.

Mastering periodicity allows one to efficiently draw the graph and extend it if necessary, as the approach is predictable thanks to the repetitive nature of trigonometric functions.