Answer:
The equation of the limacon is r = 2 - 2 sinФ ⇒ answer (a)
Step-by-step explanation:
* Lets study the limacon curve
- Equations of the form:
# r = a + b sin θ
# r = a – b sin θ
# r = a + b cos θ
# r = a – b cos θ
All will produce limacons.
* Lets examine what happens for various values of a and b.
- When the value of a is less than the value of b, the graph is
a limacon with and inner loop. ⇒ figure (1)
- When the value of a is greater than the value of b, the graph is
a dimpled limacon. ⇒ figure (2)
- When the value of a is greater than or equal to the value of 2b,
the graph is a convex limacon. ⇒ figure (3)
- When the value of a equals the value of b, the graph is a special
case of the limacon. It is called a cardioid. ⇒ figure (4)
* Notice that, in each of the graphs of the liamsons, changing
from sine to cosine does not affect the shape of the graph just its
orientation.
- Equations using sine will be symmetric to the vertical axis
- Equations using cosine are symmetric to the horizontal axis.
* The sign of b will also affect their orientation.
- The large of the graph under the x-axis or left to the y-axis b is -ve
- The large of the graph over the x-axis or right to the y-axis b is +ve
* lets look to the graph
∵ The graph is symmetric to the y- axis
∵ The graph is a cardioid
∴ a = b
∵ The large part of the graph under the x-axis
∴ b is -ve
∴ The answer is r = 2 - 2 sinФ
* The equation of the limacon is r = 2 - 2 sinФ
