The correct answer is 

The polar form of a complex number  is  where  is the modulus of the complex number and  is the angle in radians between the real axis and the line that passes through  (   and ). We can solve for  and  easily for the complex number :

which gives us

Remember that the standard form of a complex number is: , which can be rewritten in polar form as: .

To find r, we must find the length of the line  by using the Pythagorean theorem:

To find , we can use the equation 

Note that this value is in radians, NOT degrees.

Thus, the polar form of this equation can be written as 

Given these identities, first solve for  and . The polar form of a complex number is: 

 at  (because the original point, (1,1) is in Quadrant 1)

Therefore...

First, find the radius :

Then find the angle, thinking of the imaginary part as the height and the radius as the hypotenuse of a right triangle:

according to the calculator.

We can get the positive coterminal angle by adding :

The polar form is

First find the radius, :

Now find the angle, thinking of the imaginary part as the height and the radius as the hypotenuse of a right triangle:

according to the calculator.

This is an appropriate angle to stay with since this number should be in quadrant I.

The complex number in polar form is

First find :

Now find the angle. Consider the imaginary part to be the height of a right triangle with hypotenuse .

according to the calculator.

What the calculator does not know is that this angle is actually located in quadrant II, since the real part is negative and the imaginary part is positive.

To find the angle in quadrant II whose sine is also , subtract from :

The complex number in polar form is

To convert polar coordinates  to rectangular coordinates ,

Using the information given in the question, 

The rectangular coordinates are 

This is the graph of a cardiod. Based on its orientation where the cusp [pointy part] is on the y-axis, it is a sine and not cosine function. The x-intercepts are at , so the first number must be 2. Since vertically the graph goes from 0 to 4, the second number must be 2, because and .

This graph shows a rose curve with an odd number of petals.

This means that the equation will be in the form , where represents the length of the "petals" and represents the number of petals.

There are 5 petals of length 7.

This graph shows a rose curve with an even number of petals. The first petal also intersects with the x-axis. This means that the equation will be in the form where is the length of each petal, and is half the number of petals. (Note that for an odd number of petals, the rose curve will have exactly petals). In this case, the petals have length 5, and there are 8 of them [half of 8 is 4].