The graph of a limacon curve is given. without using your graphing calculator, determine which equation is correct for the graph. [-5, 5] by [-5, 5].
a. r= 2-2 sinθ
b. r= 3-2 sinθ
c. r= 2+2 sinθ
d. r= 1-3 sinθ

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

answer: the answer is 35%

the expected value of the random variable : 0, 1 and 2

step-by-step explanation: