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Chapter 8: Problem 7

Graph the polar equation \(r=1+2 \sin \theta\) for \(\theta \in[0,2 \pi]\).

Short Answer

Expert verified
Graph the equation \(r = 1 + 2 \sin \theta\) by plotting points and connecting them; the graph is a limacon with an inner loop.

Step by step solution

01

Understanding the Polar Equation

The given polar equation is \(r = 1 + 2 \sin \theta\). Here, \(r\) is the radius or the distance from the origin, and \(\theta\) is the angle measured counterclockwise from the positive x-axis. The goal is to graph this equation for \(\theta\) ranging from 0 to \(2\pi\).
02

Generate Coordinate Points

To graph this, generate a set of points by calculating \(r\) for various values of \(\theta\). Start by computing \(r\) for \(\theta = 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}, \text{etc},\) up to \(2\pi\).
03

Calculate Sample Points

Here are some sample points: - For \(\theta = 0\), \(r = 1 + 2 \sin(0) = 1\) - For \(\theta = \frac{\pi}{2}\), \(r = 1 + 2 \sin(\frac{\pi}{2}) = 3\) - For \(\theta = \pi\), \(r = 1 + 2 \sin(\pi) = 1\) - For \(\theta = \frac{3\pi}{2}\), \(r = 1 + 2 \sin(\frac{3\pi}{2}) = -1\) - Continue this process for several more values including the intermediates.
04

Plot Points on Polar Grid

Plot all the calculated coordinate points using a polar grid. Each point is represented by its corresponding \(r\) and \(\theta\).
05

Draw the Graph

Connect the plotted points smoothly to form the graph. The shape obtained will be a limacon with an inner loop due to the negative value when \(\theta = \frac{3\pi}{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polar Coordinates
Polar coordinates offer a unique way to locate points in a plane using a radius and an angle. In this system, a point is specified by its distance from a reference point (usually the origin) and the angle from a reference direction (often the positive x-axis). The configuration follows:
  • The radius, denoted by \(r\), represents the distance from the origin.
  • The angle, denoted by \(\theta\), is measured counterclockwise from the positive x-axis.
This differs from Cartesian coordinates, which use x and y values to specify locations. Polar coordinates are particularly useful for equations and shapes that exhibit rotational symmetry. They streamline the process of graphing circular and spiral patterns that would be more complex in the Cartesian system.
Graphing Polar Equations
Graphing polar equations involves plotting points based on their radius \(r\) and angle \(\theta\). For example, consider the polar equation \(r = 1 + 2 \sin \theta\). Here are steps to graph this equation:
  • Identify the function \(r\) in terms of \(\theta\).
  • Generate coordinate points by calculating \(r\) for various angles from 0 to \(2\pi \).
  • Plot these points on a polar grid where each point corresponds to its \(r\) and \(\theta\) values.
  • Connect the points smoothly to visualize the curve.
When generating points, it helps to use common angles, such as \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), and so on. This method reveals the shape and structure of the graph efficiently. Also, be mindful of negative \(r\) values, as they indicate movement in the opposite direction of the angle \(\theta\).
Limacon
A limacon is a type of polar graph named after the French word for 'snail.' It is defined by polar equations of the form \(r = a + b \sin \theta\) or \(r = a + b \cos \theta\), where \(a\) and \(b\) are constants. Limacons can have various shapes depending on the values of \(a\) and \(b\):
  • When \(a < b\), the limacon has an inner loop.
  • When \(a = b\), it forms a cardioid shape.
  • When \(a > b\), the limacon looks like a distorted circle.
In the equation \(r = 1 + 2 \sin \theta\), the limacon has an inner loop because \(a < b\). The curve loops inward when \(r\) becomes negative, specifically at \(\theta = \frac{3\pi}{2}\) where \(\sin \left(\frac{3\pi}{2}\right) = -1\), giving \(r = -1\). Understanding these properties helps in predicting and sketching the graph accurately.

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Most popular questions from this chapter

Graph the polar equation \(r=2+\sin \theta\) for \(\theta \in[0,2 \pi]\)

Graph the polar equation \(r=1+\sin (3 \theta)\) for \(\theta \in[0,2 \pi]\).

The following are described in Cartesian coordinates. Rewrite them in terms of spherical coordinates. (a) \(z=x^{2}+y^{2}\) (b) \(x^{2}-y^{2}=1\) (c) \(z^{2}+x^{2}+y^{2}=6\) (d) \(z=\sqrt{x^{2}+y^{2}}\) (e) \(y=x\) (f) \(z=x\)

The following relations are written in terms of Cartesian coordinates \((x, y) .\) Rewrite them in terms of polar coordinates, \((r, \theta)\). (a) \(y=x^{2}\) (b) \(y=2 x+6\) (c) \(x^{2}+y^{2}=4\) (d) \(x^{2}-y^{2}=1\)

Describe the surface \(r=5\) in Cartesian coordinates, where \(r\) is one of the cylindrical coordinates.
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