Graphing with polar coordinates Problem

In summary, the given equation is r = 1/2 + cos(theta), which represents a Limacon curve. The maximum value of r is 3/2 at 0 degrees, and the minimum value is -1/2 at 180 degrees. The curve is symmetrical about the x-axis. By calculating the values of r from 0 to 180 degrees, the graph can be drawn. However, at 120 degrees, there is a problem in representing both 0 and (-0.5, 180 degrees). The solution is to pretend to draw 0.5 at 180 degrees and then flip it back to positive 0.5 at 0 degrees.
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Homework Statement



Draw the graph of r = 1/2 + cos(theta)


Homework Equations



The equation is itself given in the question. It is a Limacon.


The Attempt at a Solution



Step-1 ---> Max. value of r is 1/2 + 1 = 3/2 [ at cos (0) ]
Min. value of r is 1/2 - 1 = -1/2 [ at cos (Pi) ]

Step-2 ---> Symmetry The curve is symmetrical about x-axis.

Step-3 ---> I calculated all the values of "r" from 0 to 180 degrees. These are 1.5, 1.336, 1.207, 1, 0.5, 0, -0.207, -0.336, -0.5. These are in standard form like 0, 30, 45... and so on.

Step-4 ---> Now I draw the graph. Till 90 degrees its all right; but at 120 degrees the problem comes. How can I draw show 0 as well as 120 degrees? How can I show (-0.5, 180 degrees) ?
 
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  • #2
Pretend you are drawing 0.5 at 180 degree, but since the radius is negative, flip it back to positive 0.5 at 0 degrees.
 
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  • #3
Okay, thanks a lot.
 

What are polar coordinates and how are they used in graphing?

Polar coordinates are a way of representing points on a two-dimensional plane using distance and angle measurements. In graphing, they are used to plot points and create curves or shapes by specifying a point's distance from the origin and its direction or angle from a reference line.

What is the difference between polar coordinates and Cartesian coordinates?

Cartesian coordinates, also known as rectangular coordinates, use the x and y axes to represent points on a graph. Polar coordinates use a distance measurement (r) from the origin and an angle measurement (θ) to represent points. While Cartesian coordinates are more commonly used, polar coordinates can be useful for certain types of graphs and equations.

How do you convert polar coordinates to Cartesian coordinates?

To convert polar coordinates (r, θ) to Cartesian coordinates (x, y), you can use the following formulas: x = r cos(θ) and y = r sin(θ). This will give you the x and y coordinates of the point on the Cartesian plane.

What are some common polar graphs and their equations?

Some common polar graphs include circles (r = a), cardioids (r = a + cos(θ)), and limaçons (r = a + b cos(θ)). Other more complex graphs include logarithmic spirals (r = a^θ) and rose curves (r = a cos(θ/b)).

What is the purpose of graphing with polar coordinates?

Graphing with polar coordinates can be useful for visualizing and understanding certain types of equations and curves. It can also be used in applications such as navigation and engineering, where distance and angle measurements are important. Additionally, polar coordinates provide an alternative way of representing points on a graph, which can be helpful in problem-solving and understanding mathematical concepts.

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