INTRODUCTION TO POLAR COORDINATES

Example 1:

For the points A and B in the Polar Coordinate System below, find four different sets of polar coordinates with or and r positive and negative. Express angle in degrees and radians.

The angle spacing is in 15o /12 increments!

Point A:

Note that point A lies on the intersection of the red circle and the red line. Since the polar coordinates are to contain angles between 2 and 2and 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 B:

Note that point B lies on the intersection of the green circle and the green line. Since the polar coordinates are to contain angles between 2 and 2and 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:

Example 2:

For the points C, D, and E in the Polar Coordinate System below, find four different sets of polar coordinates with and r positive and negative.  Express angle in degrees.

The angle spacing is in 15o /12 increments!

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 360o and 360o 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 360o and 360o 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 lineSince the polar coordinates are to contain angles between 360o and 360o 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 = 60o 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 60o, which equals 60o, to find this angle. Remember, negative angles and their positive counterparts have the same reference angle!

Namely, = 360o 60o = 300o

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 = 30o 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 30o, which equals 30o, to find this angle.

Namely, = 180o + 30o = 210o

The polar coordinates are .

Example 7:

Change the rectangular coordinates (3, 0) to EXACT polar coordinatesLet (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 .