Problem 38 - Algebra and Trigonometry

Question

Sketch a graph of the polar equation. $$ r^{2}=4 \sin 2 \theta $$

Step-by-Step Solution

Answer

The graph is a 4-petal rose curve, symmetric about the \( \theta = \frac{\pi}{4} \) line, extending to radius 2.

Step 1: Understand the Equation Form

Our given polar equation is \( r^{2} = 4 \sin 2\theta \). Notice that this equation is in terms of \( r^{2} \), which suggests it might correspond to a limacon or rose curve. In polar coordinates, equations involving \( \sin \) and \( \cos \) functions are commonly used for rose curves.

Step 2: Identify Symmetry and Petals

The given equation \( r^{2} = 4 \sin 2 \theta \) can be rewritten in terms of \( r = \sqrt{4 \sin 2 \theta} \) or \( r = -\sqrt{4 \sin 2 \theta} \). The \( \sin 2\theta \) part indicates potential symmetry about the line \( \theta = \frac{\pi}{4} \), and since it is of the form \( \sin n\theta \), we expect it to represent a rose curve with \( 2n = 4 \) petals.

Step 3: Determine Key Points and Ranges

To understand the behavior of \( r \), set \( 2\theta = k\pi \) where \( k \) is an integer. \( \sin k\pi = 0 \), making \( r = 0 \). Meanwhile, when \( \sin 2\theta = 1 \), \( r^{2} = 4 \), thus \( r = 2 \). This happens at \( \theta = \frac{\pi}{4}, \frac{5\pi}{4}, \frac{9\pi}{4} \), and similarly for other multiples.

Step 4: Sketch the Curve

With the information about symmetry and key values, sketch the rose curve. The 4 petals appear at angles derived from \( \theta \approx \frac{\pi}{4} + k\frac{\pi}{2} \) for \( k = 0, 1, 2, 3 \), each extending radially out to a maximum radius of 2. These configurations reflect both positive and negative square roots.

Step 5: Finalize and Label the Graph

Ensure the graph has a symmetrical appearance with petals systematically distributed around the polar coordinate system. Label important angles and radii to clearly present the structure of the rose curve.

Key Concepts

Graphing Polar Coordinates

Polar coordinates provide a unique way to represent points in a plane using a combination of distance and angles. Unlike the Cartesian coordinate system, which uses \((x, y)\) pairs, polar coordinates use \((r,\theta)\) pairs. Here, \(r\) represents the radial distance from the pole (or the origin), and \(\theta\) denotes the angle measured from the positive x-axis. To graph polar coordinates, follow a clear set of steps:

• Identify the angle \(\theta\) and the radial distance \(r\).
• Move outward from the pole in the direction of \(\theta\) and mark the point at the distance \(r\).

Repeating this process for various \((r, \theta)\) can help you graph points and shapes in the polar coordinate system. It's important to remember that some shapes like circles, loops, and roses can be easily visualized using polar graphs.

Symmetry in Polar Graphs

Understanding symmetry in polar graphs is essential for efficiently sketching and identifying these graphs. Symmetry can simplify graphing because it reduces the number of calculations needed. Polar graphs can exhibit several types of symmetry:

• Line Symmetry: A polar graph is symmetric about the line \(\theta = \frac{\pi}{2}\) if replacing \(\theta\) with \(\pi - \theta\) yields the same equation.
• Polar Symmetry: If the equation remains unchanged when replacing \(r\) with \(-r\), the graph has symmetry about the origin or pole.
• Symmetry About a Line: In the case of sinusoids like \(\sin 2\theta\), symmetry may be about lines other than the axes, such as \(\theta = \frac{\pi}{4}\).

This concept is crucial when dealing with polar equations that exhibit specific periodicity and symmetry properties, as it can guide you in predicting the shape and orientation of the graph.

Rose Curves in Polar Equations

Rose curves are a fascinating class of polar equations characterized by their distinctive petal shapes. These curves typically take the form \(r = a \sin(n\theta)\) or \(r = a \cos(n\theta)\). The parameter \(n\) determines the number of petals:

• If \(n\) is even, the number of petals is \(2n\).
• If \(n\) is odd, the number of petals is \(n\).

For example, an equation such as \(r = 2 \sin(2\theta)\) describes a rose curve with four petals. Recognizing the form of the equation allows you to anticipate the number and layout of the petals before graphing. When graphing rose curves, identify key angles where \(r\) reaches maximum values, typically where the sine or cosine function equals 1. This helps outline each petal's orientation and ensures that the rose curve has the correct structure and symmetry across the polar grid.