Question

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

Short answer

The graph is a limaçon with an inner loop and is symmetric about the line \( \theta = \frac{3\pi}{2} \).

Step-by-step solution

Identify the Polar Equation

The given polar equation is \( r = 2 - 2 \sin \theta \). This equation represents a limaçon, a common shape in polar coordinates. Our task is to understand the nature of this limaçon to sketch it accurately.

Analyze the Equation Structure

The equation is in the form \( r = a \, - \, b \sin \theta \). For this particular form, where \( a = b \), the graph is a limaçon with an inner loop. Given \( a = 2 \) and \( b = 2 \), the loop characteristic confirms the shape.

Calculate Critical Points

Determine key angles to evaluate the radius \( r \). At \( \theta = 0 \), \( r = 2 - 2\cdot0 = 2 \). At \( \theta = \frac{\pi}{2} \), \( r = 2 - 2\cdot1 = 0 \). At \( \theta = \pi \), \( r = 2 + 2\cdot0 = 2 \). At \( \theta = \frac{3\pi}{2} \), \( r = 2 + 2\cdot(-1) = 4 \). These values help in sketching the graph.

Plot Critical Points

Use polar coordinates to plot: \((0,2)\), \((\frac{\pi}{2},0)\), \((\pi,2)\), and \((\frac{3\pi}{2},4)\). These points establish the structure for the limaçon with an inner loop.

Sketch the Graph

Using the critical points, sketch the limaçon. Start at \( (0,2) \), move inward toward the origin at \( (\frac{\pi}{2},0) \), loop out to \( (\frac{3\pi}{2},4) \), and back to \( (\pi,2) \). Ensure the loop is inside the main part of the limaçon.

Identify Symmetry

Observe that the equation \( r = 2 - 2\sin\theta \) implies symmetry about the line \( \theta = \frac{3\pi}{2} \). Checking symmetry visually or through replacement techniques (replacing \( \theta \) with \( -\theta \) or \( r \) with \(-r\) aids confirmation.

Key concepts

Limaçon

In polar coordinates, a limaçon is a fascinating and distinct curve. Its shape relies on the values in the polar equation, usually given as \( r = a \, \pm \, b \sin \theta \) or \( r = a \, \pm \, b \cos \theta \). The graph takes on unique characteristics depending on the relation between \(a\) and \(b\).

When \(a = b\), as in our equation \( r = 2 - 2 \sin \theta \), the limaçon forms with an inner loop. This loop appears because the equation structure dictates negative radii values for certain angles, indicating points are closer to the center than other parts of the curve.

A limaçon can exhibit several appearances:

• With a loop - Occurs when \(a = b\) or \(a < b\). The inner loop is a captivating part of the graph.
• Cardioid - When \(a = b\), the limaçon touches the origin without an inner loop.
• Without a loop - When \(a > b\), the curve appears more like a dimpled or dented heart.

Recognizing these features helps in understanding and predicting the graph's behavior.

Symmetry in Polar Graphs

Symmetry plays a crucial role in graphing polar equations. Symmetrical properties help simplify calculations and graphing procedures by reducing repetitive evaluations.

For the equation \( r = 2 - 2 \sin \theta \), symmetry can be identified about specific lines:

• Vertical Symmetry - Often checked around the initial line of sight, such as \( \theta = \frac{\pi}{2} \).
• Horizontal Symmetry - Observed when the graph reflects over a horizontal line, such as the polar axis.
• Origin Symmetry - The graph can reflect through the pole, noting when the angle \( \theta \) and radius \( r \) are both reversed.

In this particular limaçon, there's clear symmetry about the line \( \theta = \frac{3\pi}{2} \). By checking multiple points or using transformation rules, we verify that the equations produce symmetrical reflections.

Such symmetry helps when sketching the complete graph and assures us of balanced and accurate representations.

Graphing Polar Equations

Graphing polar equations requires understanding both the polar coordinate system and the specific nature of each equation.

To graph a polar equation like \( r = 2 - 2 \sin \theta \), follow a logical sequence:

• Identify key points - Calculate radii for angles such as \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \).
• Plot the points - Using the calculated radii, plot each point on polar grid paper, noting direction and distance from the pole (origin).
• Connect the points - Establish the graph by smoothly connecting plotted points, observing any loops or cusps that form.

Connecting the points must respect the calculated radii, and the graph should be symmetric where applicable.

Lastly, confirming the sketch against known templates of limaçons or other curves can enhance accuracy. With practice, graphing polar equations becomes a straightforward task, unlocking a new dimension of mathematical beauty.