Problem 20 Sketch the graph of each polar e... [FREE SOLUTION]

Chapter 6: Problem 20

Sketch the graph of each polar equation. $$r=\frac{8}{2-4 \cos \theta}$$

Short Answer

Expert verified

The graph of the given polar equation, $ r=\frac{8}{2-4 \cos \theta} $, forms a lima莽on, looping downwards due to the negative coefficient of cosine in the given equation, with key points at (蟺/2, 4), (3蟺/2, 4), (0, -鈭�), and (蟺, 4).

Step by step solution

Simplify the Equation

First, simplify the given equation: $ r=\frac{8}{2-4 \cos \theta} = \frac{8}{2(1-2\cos \theta)} = \frac{4}{1-2\cos \theta} $. This form of equation represents a lima莽on, a type of polar graph.

Identify properties of 尝颈尘补莽辞苍

From the equation, identify $ k = -2$ as a non-positive constant multiplied with cos(胃), and $ a = 4 $ as the constant. The negative value of cosine suggests that the lima莽on will be 'flipped' or mirrored along the polar axis.

Plot the key points

Key points occur when $ \cos \theta = 0, 1, -1$. These include: (胃, r) = (蟺/2, 4), (3蟺/2, 4), (0, - 鈭�), and (蟺, 4). Plot these on the polar grid.

Draw the lima莽on

Connect the points by smoothly drawing loop opening downward towards negative axis due to negative coefficient of cosine, this takes into account the nature of a lima莽on graph.

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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

Understanding a lima莽on is crucial to sketching the graph of polar equations like the one given. A lima莽on is a type of polar graph that showcases distinct features based on the cosine or sine functions and their coefficients.
There are different forms of lima莽ons:

Simple lima莽on (with inner loop).
Cardioid (heart-shaped lima莽on).
Dimpled lima莽on (without inner loop).
Convex lima莽on (without dimples or loops).

In our equation, when simplified, the equation resembles the form of a lima莽on with an inner loop because the modulus of the constant with cosine, $ |-2| $, is greater than 1.
This means the graph will cover more area and might intersect itself, causing an inner loop.

Polar Graph

A polar graph is a way to represent complex equations visually using polar coordinates. Unlike Cartesian graphs, which use $x$- and $y$-coordinates, polar graphs use $r$ (radius) and $\theta$ (angle) to plot points.
This type of graph is handy when dealing with circular and spiral patterns or any situation where direction and magnitude (rather than horizontal and vertical measures) matter.
In the context of our problem, the polar graph helps visualize the lima莽on by plotting points in a circular grid based on the $\theta$ angle and corresponding $r$ radius.

Plotting Polar Coordinates

Plotting polar coordinates involves determining and plotting points based on an angle $\theta$ and distance from the origin. In this exercise, the equation requires us to find points based on various values of $ \theta $ to see how $ r $ changes.
Key points in plotting might include where $ \cos \theta = 0, 1, \text{and} -1 $. These points are crucial as they determine how the polar graph will look. For example:

When $ \cos \theta = 1 $, the graph gives situations like (0, -鈭�), which means the radius goes very far in one direction.
When $ \cos \theta = 0 $, points like $(\pi/2, 4)$ are plotted.
For $ \cos \theta = -1 $, points such as $(\pi, 4)$ appear, indicating how the graph loops inward.

Each plotted point contributes to the overall shape and uniqueness of the lima莽on on the polar grid.

Trigonometric Functions

Trigonometric functions like sine and cosine play an integral role in defining and graphing polar equations. In the context of the provided lima莽on equation, the function $ \cos \theta $ modifies the radius $ r $ as $ \theta $ changes.
These functions help predict the graph's behavior. Specifically, the negative multiple $-2 \cos \theta $ in our equation suggests the graph will be flipped or mirrored compared to a regular lima莽on.

The zeroes of $ \cos \theta $ alter the graph's turning points.
The peaks ($ \cos \theta = 1 $) affect the range of $ r $.
Negative values ($ \cos \theta = -1 $) introduce loops or inversions in the graph.

By observing how $ \cos \theta $ behaves over intervals, one can predict and accurately sketch the polar graph of a lima莽on.

91影视

Sketch the graph of each polar equation. $$r=\frac{8}{2-4 \cos \theta}$$

Short Answer

Step by step solution

Simplify the Equation

Identify properties of 尝颈尘补莽辞苍

Plot the key points

Draw the lima莽on

Key Concepts

尝颈尘补莽辞苍

Polar Graph

Plotting Polar Coordinates

Trigonometric Functions

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