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Chapter 11: Problem 10

Graph the following equations. $$ r=\frac{3}{2+\sin (\theta)} $$

Short Answer

Expert verified
The graph is a dimpled lima莽on symmetric about the vertical axis.

Step by step solution

01

Identify the Equation Type

The polar equation given is in the form \( r = \frac{k}{a + b \sin(\theta)} \). This form represents a conic section, specifically a lima莽on, which is a type of polar graph with distinctive loops or dimples depending on the parameters.
02

Determine 尝颈尘补莽辞苍 Characteristics

For the equation \( r = \frac{3}{2 + \sin(\theta)} \), we identify the constants: \( k = 3 \), \( a = 2 \), and \( b = 1 \). Since \( a > b \), the graph will be a dimpled lima莽on without an inner loop.
03

Find Symmetry and Key Angles

Because the function has a \( \sin(\theta) \) component, the graph is symmetric about the vertical axis (胃 = 蟺/2). Evaluate the equation at key angles: \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \) to understand the graph's key distances from the origin.
04

Calculate Polar Coordinates

Use the polar equation to calculate specific points:- \( \theta = 0 \): \( r = \frac{3}{2 + 0} = \frac{3}{2} \)- \( \theta = \frac{\pi}{2} \): \( r = \frac{3}{2 + 1} = 1 \)- \( \theta = \pi \): \( r = \frac{3}{2 - 0} = \frac{3}{2} \)- \( \theta = \frac{3\pi}{2} \): \( r = \frac{3}{2 - 1} = 3 \)
05

Plot the Graph

Using the calculated points, plot them in polar coordinates. Join the points smoothly to reflect the dimpled characteristic of the lima莽on. The graph will have a larger radius at \( \theta = \frac{3\pi}{2} \) and symmetric dimpled shape about the vertical axis.

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Key Concepts

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

尝颈尘补莽辞苍
A lima莽on is a fascinating type of graph in polar coordinates. It appears when you graph polar equations in the form \( r = \frac{k}{a + b \sin(\theta)} \) or \( r = \frac{k}{a + b \cos(\theta)} \). The shape of a lima莽on can vary widely:
  • It can have a loop (when \( b > a \))
  • Be dimpled, resembling a heart-like shape (when \( a > b \))
  • Or form a perfect circle in some cases
In our given equation \( r = \frac{3}{2 + \sin(\theta)} \), the shape will be dimpled, as \( a \) is greater than \( b \). 尝颈尘补莽辞苍s can be playful on the graph, changing their appearance dramatically with different parameters.
Conic Sections
Conic sections are shapes created by slicing a cone with a plane. The most common conic sections include circles, ellipses, parabolas, and hyperbolas. In polar coordinates, these sections appear when equations take specific forms, predicting beautiful and varied shapes.
  • Circles have the simplest form.
  • Ellipses are stretched circles.
  • Parabolas open in one direction.
  • Hyperbolas open in two directions.
However, the lima莽on is a special type of conic section derived from speficic polar equations related to hyperbolas and ellipses. They display unique traits like dimples or loops, adding to the interesting patterns of symmetry and design in this category.
Graphing Polar Equations
Graphing polar equations starts with converting equations into polar form, like our given equation \( r = \frac{3}{2 + \sin(\theta)} \). Polar graphs are plotted using angles \( \theta \) and radius \( r \).
  • Identify the form: Check if it's a known shape like a lima莽on.
  • Analyze the constants to determine the key graph characteristics.
  • Calculate key points by substituting angles \( \theta \) to find corresponding \( r \) values.
For our lima莽on, calculate coordinates at crucial angles like \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \). Map these points into the polar plane where each set of \( (\theta, r) \) represents a unique location, creating the renowned dimpled shape.
Symmetry in Polar Graphs
Symmetry plays a critical role in understanding and sketching polar graphs effectively. It simplifies the graphing process, providing a balanced design.
  • Analyze symmetry in our equation which involves \( \sin(\theta) \), implying vertical axis symmetry (along \( \theta = \frac{\pi}{2} \)).
  • Symmetric properties reduce computation by confirming that behavior on one side reflects that of another.
  • This helps in rapidly plotting and checking your graph shape.
By understanding symmetry, you predict how the graph unfolds around its axes. With this knowledge, connect points to see how traits like dimples and loops are reflected across the axis.

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Most popular questions from this chapter

In Exercises \(1-20\), use the pair of vectors \(\vec{v}\) and \(\vec{w}\) to find the following quantities. $$ \vec{v}=\left\langle\frac{1}{2},-\frac{\sqrt{3}}{2}\right\rangle \text { and } \vec{w}=\left\langle\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right\rangle $$

In Exercises \(25-39\), find a parametric description for the given oriented curve. the triangle with vertices \((0,0),(3,0),(0,4)\), oriented counter-clockwise (Shift the parameter so \(t=0\) corresponds to \((0,0) .)\)

For Exercises \(65 a\) and 65 b below, let \(f(\theta)=\cos (\theta)\) and \(g(\theta)=2-\sin (\theta)\). (a) Using your graphing calculator, compare the graph of \(r=f(\theta)\) to each of the graphs of \(r=f\left(\theta+\frac{\pi}{4}\right), r=f\left(\theta+\frac{3 \pi}{4}\right), r=f\left(\theta-\frac{\pi}{4}\right)\) and \(r=f\left(\theta-\frac{3 \pi}{4}\right)\). Repeat this process for \(g(\theta)\). In general, how do you think the graph of \(r=f(\theta+\alpha)\) compares with the graph of \(r=f(\theta)\) ? (b) Using your graphing calculator, compare the graph of \(r=f(\theta)\) to each of the graphs of \(r=2 f(\theta), r=\frac{1}{2} f(\theta), r=-f(\theta)\) and \(r=-3 f(\theta)\). Repeat this process for \(g(\theta) .\) In general, how do you think the graph of \(r=k \cdot f(\theta)\) compares with the graph of \(r=f(\theta) ?\) (Does it matter if \(k>0\) or \(k<0 ?)\)

Convert the equation from polar coordinates into rectangular coordinates. $$ r=1+\sin (\theta) $$

In Exercises \(1-20\), plot the set of parametric equations by hand. Be sure to indicate the orientation imparted on the curve by the parametrization. $$ \left\\{\begin{array}{l} x=\cos (t) \\ y=\sin (t) \end{array} \text { for }-\frac{\pi}{2} \leq t \leq \frac{\pi}{2}\right. $$
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