Polar Coordinates & Graphs

Graph:

• (3, 45^o)

Draw a circle of radius 3 centered at the origin and mark the point where the 45^o angle intersects the circle.
• (-2, -30^o)

Draw a circle of radius 2 centered at the origin.  The negative sign indicates direction along the radius and simply means move along the radius opposite the positive direction.  Positive r's mean move toward the angle named.  Negative r's mean move away from the angle named.  Find the -30^o position along this circle and, from the origin, move 2 units away from the -30^o angle.  Mark that point.
Another way to name this point is (2, 150^o).  There are other ways to name it.

The rectangular coordinates of a point are denoted by (x,y) on a rectangular coordinate system ( the one you're used to using).

The polar coordinates of a point are denoted by (r, 0) on a polar coordinate system.  The origin of this system is called the pole.

To convert from polar coordinates to rectangular coordinates, use

• x = r cos 0
• y = r sin 0
Example:
• Change (2, -40^o) to rectangular coordinates.

 

 

x = r cos 0 = 2 cos (-40o) = 1.532
y = r sin 0 = 2 sin (-40o) = -1.286

so the rectangular coordinates for this point are (1.532, -1.286).

Use the formulas below:
• x² + y² = r²
• 0 = Arctan y/x
Example:

Change (-3,3) to polar coordinates:

• x² + y² = r²                0 = Arctan y/x

9 + 9 = r²                    0 = Arctan (3/-3)
18 = r²                         0 = Arctan (-1)
3\/2 = r                  0 = 135^o  since the point (-3, 3) is in quadrant II.

so the polar coordinates are (3\/2, 135^o).

• Express r = 5 cos 0 in rectangular form.
• Multiply both sides by r:              r² = 5 r cos 0
• Since x² + y² = r²,      then  x² + y² = 5 r cos 0
• And since x = r cos 0, then  x² + y² = 5x
• Express y = 6 in polar form.
• Since  y = r sin 0,    then   r sin 0 = 6

• r = 3
• r = 2 - cos 0
• r = cos (30)

Problems (there are no problems here yet)

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