Write the equations for two limacons (and if possible graph), (one sine and one cosine) with the same a and b values. (A does not have to equal b). You may choose to use cardioids or inner loop, dimpled, or convex limacons.
Why not use Google to find out what a limacon looks like?
This will lead you to properties and equations of such curves.
you might want to play around on wolframalpha.com by varying the parameters. You have the equations for sample curves, as well as examples in your book and online.
You know the general form is r = b + a cos?. And of course, since sin? = cos(?/2 - ?) the two are just rotations of each other.
For example,
http://www.wolframalpha.com/input/?i=r+%3D+2%2B3cos%CE%B8
Sure! Let's start by defining the equations for two limacons with the same values of a and b, one using the sine function and the other using the cosine function.
1. Sine Limacon:
The equation of a sine limacon is given by:
x = (a + b * sin(t)) * cos(t)
y = (a + b * sin(t)) * sin(t)
where t is the parameter that ranges from 0 to 2π.
2. Cosine Limacon:
The equation of a cosine limacon is given by:
x = (a + b * cos(t)) * cos(t)
y = (a + b * cos(t)) * sin(t)
where t is the parameter that ranges from 0 to 2π.
To graph these limacons, you can substitute different values of t in the equations and plot corresponding (x, y) points. Since the limacon is defined for t in the range 0 to 2π, you can choose several values of t within this range, calculate the corresponding (x, y) coordinates, and connect them to form the graph.
To find the equations for two limacons with the same values of a and b, let's start by defining some terms:
- a: The distance from the center to the "inner loop" of the limacon.
- b: The distance from the center to the "outer loop" of the limacon.
Now, let's consider the sine limacon. The equation for a sine limacon is given by:
r = a + b*sin(θ)
where r represents the distance from the origin to a point on the limacon, θ is the angle measured counterclockwise from the positive x-axis to the point, and a and b are the given values.
To graph this equation, you can use a polar coordinate system. Plot the points (r, θ) by evaluating the equation for different values of θ. Connect these points to form the graph of the sine limacon.
Now, let's consider the cosine limacon. The equation for a cosine limacon is given by:
r = a + b*cos(θ)
This equation is similar to the sine limacon, but it involves the cosine function instead. Again, use a polar coordinate system to graph this equation and connect the points (r, θ).
Remember that the shape of the limacon depends on the values of a and b. For example, if a > b, you will have a convex limacon. If a = b, you will have a cardioid limacon. If a < b, you will have an inner loop, dimpled limacon. Feel free to experiment with different values of a and b to get different shapes.
Graphing the limacons will help you visualize the shape and understand how the values of a and b affect the curves.