Problem 5 Sketch the graph of each equatio... [FREE SOLUTION]

Chapter 8: Problem 5

Sketch the graph of each equation by making a table using values of \(\theta\) that are multiples of \(45^{\circ}\).\(r=6 \cos \theta\)

Short Answer

Expert verified

Plot points for \( r = 6 \cos \theta \) using \( \theta \) multiples of \( 45^{\circ} \), then connect them to form a limacon.

Step by step solution

Understanding the Equation

The equation given is in polar form, which relates the radius \( r \) with the angle \( \theta \). It indicates how the distance from the origin varies with \( \theta \). This is a polar equation of a limacon.

Select \( \theta \) Values

Choose values for \( \theta \) that are multiples of \( 45^{\circ} \). These will be: \( 0^{\circ}, 45^{\circ}, 90^{\circ}, 135^{\circ}, 180^{\circ}, 225^{\circ}, 270^{\circ}, 315^{\circ}, 360^{\circ} \).

Calculate \( r \) for Each \( \theta \) Value

For each angle, calculate \( r = 6 \cos \theta \). Make sure to convert any angle measurements into radians if using a calculator (e.g., \( 45^{\circ} = \frac{\pi}{4} \)).

Compile the Table

Create a table with two columns: one for \( \theta \) and the other for \( r \). Fill in the values of \( r \) corresponding to each \( \theta \). For example: \( \theta = 0^{\circ} \), \( r = 6 \); \( \theta = 90^{\circ} \), \( r = 0 \); etc.

Plot the Points on Polar Graph

Plot each pair of values (\( \theta, r \)) on a polar coordinate graph. Each point's location is determined by \( \theta \) (the direction angle) and \( r \) (the distance from the origin along that angle).

Sketch the Graph

Connect the plotted points to sketch the curve. This graph should show the symmetry and shape typical of a limacon with a loop, extending from \( 6 \) units out at \( \theta = 0^{\circ} \) to forming a smaller loop around the pole.

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Key Concepts

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

Graphing Polar Equations

Polar coordinates allow us to describe points on a plane using a combination of radius and angle, rather than Cartesian coordinates which use x and y. In the polar system:

Each point is defined by a distance from the origin, called the radius (\( r \)), and an angle (\( \theta \)) from the positive x-axis.
This system is particularly useful for equations describing circular and spiral-like shapes, which we call polar equations.

Let's take an example: the polar equation \( r = 6 \cos \theta \). To graph this:

Choose angles \( \theta \) often in multiples of \( 45^{\circ} \); these are easy to compute in trigonometric terms and translate well into radians.
For each angle, compute the radius (\( r \)), allowing you to plot points on a polar graph paper.

As you plot these points and connect them, the graph will reveal the pattern described by the equation.

Limacon

The graph of \( r = 6 \cos \theta \) illustrates a unique type of curve known as a limacon. Limacons are special because they can take on different forms based on certain parameters in their equations. Here are some characteristics:

When the cosine function is involved, like in our equation, the limacon is oriented horizontally.
Limacons may have inner loops, a dimple, or they may appear to be just a cardioid shape, depending largely on the coefficients in the equation.
For the equation \( r = 6 \cos \theta \), the graph forms a limacon with an inner loop. The loop is visible because the radius (\( r \)) can become negative, pulling the loop inside and crossing the pole.

Understanding these shapes helps in predicting the behavior of the graph when changes are made to the function.

Trigonometric Functions

Trigonometric functions, like sine and cosine, play a crucial role in polar equations. They are periodic and oscillate between a maximum and minimum value, which influences the radius (\( r \)) in polar graphs.Consider \( r = 6 \cos \theta \):

The cosine function varies between -1 and 1, meaning \( r \) will range from \(-6\) to \(6\) as \( \theta \) changes.
At \( \theta = 0^{\circ} \) and \( \theta = 360^{\circ} \), \( \cos \theta \) is 1, giving the maximum radius of 6.
Meanwhile, at \( \theta = 90^{\circ} \) and \( \theta = 270^{\circ} \), \( \cos \theta \) is 0, providing no radius, or zero distance from the origin.

These trigonometric properties create the repetitive and symmetrical patterns seen in polar graphs, which make polar equations fascinating and visually intriguing.

91影视

Sketch the graph of each equation by making a table using values of \(\theta\) that are multiples of \(45^{\circ}\).\(r=6 \cos \theta\)

Short Answer

Step by step solution

Understanding the Equation

Select \( \theta \) Values

Calculate \( r \) for Each \( \theta \) Value

Compile the Table

Plot the Points on Polar Graph

Sketch the Graph

Key Concepts

Graphing Polar Equations

Limacon

Trigonometric Functions

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