Question

Sketch a graph of the polar equation and identify any symmetry. $$ r=3 \sin (2 \theta) $$

Short answer

The graph is a 4-petal rose curve, symmetric about the y-axis.

Step-by-step solution

Understand the Equation

The given polar equation is \( r = 3 \sin(2\theta) \). This represents a polar curve, specifically a rose curve with a length of each petal determined by the coefficient 3 and the number of petals determined by the coefficient of \(\theta\), which is 2.

Determine Number of Petals

Since the equation is of the form \( r = a \sin(n\theta)\), where \( n \) is even, the number of petals in the rose curve is \( 2n \). Therefore, for \( n = 2 \), the rose will have \( 4 \) petals.

Check for Symmetry

This polar equation can be checked for symmetry. Since the function involves \( \sin \), we look for symmetry about the line \( \theta = \frac{\pi}{2} \). Verifying, substituting \( \theta = \pi - \theta \) shows the function remains itself. Hence, the graph is symmetric about the y-axis.

Plot Key Points

Start by substituting key values of \( \theta \) to understand how \( r \) changes. For example, when \( \theta = 0\), \( r = 0 \); when \( \theta = \frac{\pi}{4} \), \( r = 3\sin(\frac{\pi}{2}) = 3 \); and when \( \theta = \frac{\pi}{2} \), \( r = 0 \). Continue these points until a full cycle is completed at \( 2\pi \).

Sketch the Graph

Using the key points and symmetry findings, sketch the graph in polar coordinates. Each petal extends from the origin and has maxima at multiples of \( \frac{\pi}{4} \) radians, coinciding with 45-degree increments to account for \( 4 \) petals.

Key concepts

Rose Curves

Rose curves are a fascinating type of polar graph characterized by petal-like structures. The general equation for a rose curve is either \( r = a \sin(n\theta) \) or \( r = a \cos(n\theta) \). Here, \( a \) determines the length of each petal, while \( n \) determines the number of petals. A key point to remember is that whether \( n \) is odd or even changes the number of petals:

• If \( n \) is odd, the rose curve has \( n \) petals.
• If \( n \) is even, it boasts \( 2n \) petals.

In the case of our exercise, given \( r = 3 \sin(2\theta) \), results in a rose curve with four petals. Each petal extends from the origin, sweeping through angles and forming a symmetry that's both aesthetic and mathematically intriguing.
Such curves provide a perfect blend of simplicity and complexity. They can be deeply explored by understanding these coefficients which greatly influence their shape and structure. These petal patterns are often found in nature and art, making them not just mathematical constructs but also sources of aesthetic inspiration.

Symmetry in Polar Graphs

Symmetry is a crucial concept when working with polar graphs, as it helps simplify graphing tasks and ensures accuracy. Polar graphs can exhibit symmetry in different ways:

• About the x-axis
• About the y-axis
• About the origin

For our example, \( r = 3\sin(2\theta) \), we're particularly interested in symmetry about the y-axis, as indicated by the sine function. To test this symmetry, you can substitute \( \theta \) with \( \pi - \theta \) in the equation and see if it remains unchanged.
Successfully doing so confirms that the graph is symmetric about the y-axis, meaning it mirrors itself across the vertical plane. This symmetry is helpful not only in plotting but also in analyzing the behavior and structure of the curve. Understanding symmetry simplifies the graph drawing process and strengthens our grasp of polar functions.

Graphing Polar Equations

Graphing polar equations is a unique process that differs from the conventional Cartesian coordinate system. When plotting polar graphs, each point is represented by a distance \( r \) from the origin and an angle \( \theta \) from the polar axis (often equivalent to the positive x-axis).
To effectively graph a polar equation like \( r = 3 \sin(2\theta) \), follow these steps:

• Identify the type of polar curve (e.g., rose curves, spirals).
• Determine any symmetry, which aids in simplifying the graphing task.
• Identify key points by choosing various \( \theta \) values and calculating their corresponding \( r \).

Consider plotting points: for instance, at \( \theta = 0, \frac{\pi}{4}, \frac{\pi}{2} \), compute \( r \) to find positions on the graph. Repeat for a full cycle to ensure completion of all petals.
Graphing these intriguing curves is both an art and a science, requiring careful plotting of points and an understanding of mathematical properties to create beautiful, symmetrical designs that reflect the specified polar equation.