Point C: 

Note that point C lies on the intersection of the blue circle and the blue line. Since the polar coordinates are to contain angles between 360^o and 360^o and positive and negative r-values, we find the following coordinates:

r positive and positive:

r positive and negative:

r negative and positive:

r negative and negative:

Point D: 

Note that point D lies on the intersection of the blue circle and the black line. Since the polar coordinates are to contain angles between 360^o and 360^o and positive and negative r-values, we find the following coordinates:

r positive and positive:

r positive and negative:

r negative and positive: 

r negative and negative:

Point E: 

Note that point E lies on the intersection of the orange circle and the orange line. Since the polar coordinates are to contain angles between 360^o and 360^o and r  values that are positive and negative, we find the following coordinates:

r positive and positive:

r positive and negative:

r negative and positive: 

r negative and negative:

Example 3:

Change the polar coordinates to EXACT rectangular coordinates. 

NOTE:  It is BEST to plot the point in the Polar Coordinate System first before its conversion to rectangular coordinates.

We must use the conversions and !!!

We note that the point lies in Quadrant I of the Rectangular Coordinate System. Therefore, the x-variable and the y-variable are positive!

NOTE: You either must have memorized that or you must have MATHPRINT on your calculator!

The rectangular coordinates are

Example 4:

Change the polar coordinates to EXACT rectangular coordinates. 

NOTE:  It is BEST to plot the point in the Polar Coordinate System first before its conversion to rectangular coordinates.

We must use the conversions and !!!

We note that the point lies in Quadrant IV of the Rectangular Coordinate System.  Therefore, the x-variable is positive and the y-variable negative!

NOTE: You either must have memorized that or you must have MATHPRINT on your calculator!

The rectangular coordinates are .

Example 5:

Change the rectangular coordinates to EXACT polar coordinates. Let (positive r-value) and (positive angle). Express the angle in degrees.

NOTE:  It is BEST to plot the point in the Rectangular Coordinate System first before its conversion to polar coordinates.

We must use the conversions and  !!! 

r-value:  

Knowing that , we can find

and

However, given the restriction , we will only keep .

Angle :

Knowing that , we can find

Using the calculator, we find .

Is = 60^o the correct angle?

Given the restriction we must find a positive angle.

Knowing the location of the point in the Rectangular Coordinate System, we realize that we must find a QIV angle.

We will use the Reference Angle of 60^o, which equals 60^o, to find this angle. Remember, negative angles and their positive counterparts have the same reference angle!

Namely, = 360^o 60^o = 300^o

 The polar coordinates are .

Example 6:

Change the rectangular coordinates to EXACT polar coordinates. Let (positive r-value) and (positive angle).  Express the angle in degrees.

NOTE:  It is BEST to plot the point in the Rectangular Coordinate System first before its conversion to polar coordinates.

We must use the conversions and  !!! 

r-value:

then

However, given the restriction , we will only keep .

Angle :

Using the calculator, we find .

Is = 30^o the correct angle?

Knowing the location of the point in the Rectangular Coordinate System, we realize that we must find a QIII angle.

We will use the Reference Angle of 30^o, which equals 30^o, to find this angle.

Namely, = 180^o + 30^o = 210^o

 The polar coordinates are .

Example 7:

Change the rectangular coordinates (3, 0) to EXACT polar coordinates. Let (positive r-value) and (positive angle). Express the angle in radians.

NOTE:  It is BEST to plot the point in the Rectangular Coordinate System first before its conversion to polar coordinates.

We must use the conversions and  !!! 

r-value:

then

However, given the restriction , we will only keep .

Angle :

Using the calculator, we find .

Is = 0 the correct angle?

Since is a quadrantal angle, we are going to show the following picture illustrating the tangent ratio and its numeric values for quadrantal angles.

Knowing the location of the point in the Rectangular Coordinate System, we realize that the only possibility for the angle is the value . 

The polar coordinates are .

Example 8:

Change the rectangular coordinates (0, 2) to EXACT polar coordinates. Let (positive r-value) and (positive angle).  Express the angle in radians.

NOTE:  It is BEST to plot the point in the Rectangular Coordinate System first before its conversion to polar coordinates.

We must use the conversions and  !!! 

r-value:

then

However, given the restriction , we will only keep .

Angle :

We know that the tangent ratio is only undefined given quadrantal angles. Therefore, are going to show the following picture illustrating the tangent ratio and its numeric values for quadrantal angles.

Knowing the location of the point in the Rectangular Coordinate System, we realize that the only possibility for the angle is the value /2. 

The polar coordinates are .