Let a and b be real numbers. Describe the graph of r = a cos (theta + b) as much as possible. (The intent of this question is to get as specific a description as possible that holds for all possible values of a and b.)
How would I proceed on this problem? I know r = cos theta is a plot of a circle
but how would I insert a and b in there? Thanks
Have some fun here, changing the values of a and b
I used a=4 and b=3
http://www.wolframalpha.com/input/?i=plot+r+%3D4cos%28Ø%2B3%29+%2C-6.28+%3C+Ø+%3C+6.28
Notice I also placed a domain on Ø from -2π to +2π
things really get interesting if you put a coefficient in front of the Ø , such as
r = 4cos(7Ø + 3)
cos(θ+b) just rotates the figure around the origin by b.
In effect, it rotates the x- and y-axes by b.
acos(θ+b) scales the rotated figure by a factor of a.
To describe the graph of the polar equation r = a cos(theta + b), we can break down the components and understand how they affect the shape.
1. a represents the amplitude: The value of a determines the maximum distance of the graph from the origin. If a is positive, the graph will oscillate above and below the x-axis. If a is negative, the graph will reflect across the x-axis.
2. cos(theta + b) represents the argument: The term inside the cosine function, (theta + b), introduces a phase shift by adding an angle b to theta. The value of b determines the direction and amount of the phase shift. For positive b, the graph moves in the counterclockwise direction, while for negative b, the graph moves in the clockwise direction.
Combining these two components, we can describe the graph as follows:
- If a is positive, the graph will comprise a series of "petals" that oscillate above and below the x-axis, similar to a rose curve, with a maximum distance from the origin determined by the value of a.
- If a is negative, the graph will be a reflection of the positive a graph across the x-axis.
- The number of petals will depend on the relationship between a and b. For example, if a = 1 and b = pi/3, there will be three petals in a counterclockwise direction. If b is negative, the petals will appear in a clockwise direction.
To plot the graph, start by varying theta from 0 to 2pi and evaluate r for each value of theta. Then, convert the polar coordinates (r, theta) to Cartesian coordinates (x, y) using the relationships:
x = r * cos(theta)
y = r * sin(theta)
By substituting the computed values of r and theta, you'll obtain a set of (x, y) points that represent the polar graph. Plotting these points will give you a clearer visual representation of the graph.
Note: The graph may look different for specific values of a and b, so it's recommended to experiment with different combinations to gain a better understanding of the effects of a and b on the polar equation.