Notes 10.7 – Polar Coordinates Rectangular Grid (Cartesian Coordinates) Polar Grid (Polar Coordinates) Origin Pole Positive X-axis Polar axis There is no equivalent to the y-axis in polar coordinates.
Notes 10.7 – Polar Coordinates Rectangular Grid (Cartesian Coordinates) Polar Grid (Polar Coordinates) Point (x, y) Point (r, ) x = Horizontal Distance The point doesn’t move. The point doesn’t move. We are just changing how the point is labeled. y = Vertical Distance r = Radial Distance (outward from pole) = Angle from polar axis (measured counterclockwise)
00 The angles, in degrees……… 90 180 270 , 360 45 30 60 75 15 Polar graphs can come in different increments. For this one, there are 6 steps to get to the 90 location so each step would be 90/6 or 15 105 120 165 150 135 225 210 240 255 195 285 300 345 330 315
The angles, in radians……… 0 , 2 Polar graphs can come in different increments. For this one, there are 6 steps to get to the /2 location so each step would be ( /2)/6 or /12 Since the next one would be 6 /12 which is /2, we should feel comfortable with our counting.
If you kept going, and reduced your fractions, you would have……
Ex. 1) Plot the point Find /4. Here it is!! Go out 4 from the pole and plot the point.
Alternate labels…… This point could stay in the same place but be labeled differently. If I go completely around the circle, the point would be If I go around the circle the other way, the point would be
Alternate labels…… You can also look at it this way…... If I go out on the 5 /4 line but go in a negative direction, the point would be The point never moved, it was just labeled differently. In rectangular coordinates, there is not this issue. There is only one way to label each point. In polar, we can label a point an infinite number of ways. Fortunately, MOST of the time, we only look at values of between 0 and 2 .
Ex 2) Plot the following points: D C B A
Converting from Polar to Rectangular Coordinates Remember from your basic trig: x = r cos y = r sin
Ex 3) Convert from Polar to Rectangular Coordinates So the rectangular coordinates of the point would be:
The point doesn’t move…… Take the point from part a of the previous example and plot it on the polar grid. Note where the pole is. If we bring in the rectangular grid so that the pole & origin line up (as they should)….. Then you see the point on the rectangular grid in the same location. …at least as well as your teacher can line up the grids!!
Converting from Rectangular to Polar Coordinates Remember from your basic trig: In conjunction with the Pythagorean Theorem: Use these to convert (x, y) to (r, ).
Ex 4) Convert the following from Rectangular to Polar Coordinates a) (0, 3) Plot the point on a polar grid as if it were rectangular. You will see that the polar coordinates are: b) (2, -2) So the polar coordinates would be: But that angle would be in the 4 th quadrant and our point is in the 2 nd quadrant. Thus we must add . So the polar coordinates would be:
Steps for Converting from Rectangular to Polar Coordinates 1.Always plot the (x, y) point. 2.Find r using: 3.Find using: If x, y is in Quadrant I, is OK. If x, y is in Quadrant II or III, add to get in the correct location. If x, y is in Quadrant IV, add 2 to get in the correct location and between 0 and 2 .
Transforming Equations Polar to Rectangular Ex 5) Transform r = 6 cos from polar to rectangular form. r = 6 cos Since our conversions don’t involve r or just cos but rather r 2 and r cos , multiply both sides by r to get the correct format. r 2 = 6r cos Now use the same conversions as earlier with points. r 2 = 6r cos x 2 + y 2 = 6x
Ex 5) Continued…… Since this is the equation of a circle, we can convert to standard form. Center (3, 0) with a radius of 3. (x - 3) 2 + y 2 = 9 x 2 + y 2 = 6x x 2 – 6x + y 2 = 0 x 2 – 6x y 2 = 0 + 9
Transforming Equations Rectangular to Polar Ex 6) Convert 4xy = 1 to a polar equation. Solve for r or r 2, if possible. Use the same conversions as earlier with points. 4xy = 1 4(r cos )(r sin ) = 1 4r 2 cos sin = 1 2r 2 sin 2 = 1 r 2 = ½ csc 2