REVIEW 9.1-9.4 Pola Coodinates and Equations
You ae familia with plotting with a ectangula coodinate system. We ae going to look at a new coodinate system called the pola coodinate system.
The cente of the gaph is called the pole. Angles ae measued fom the positive x axis. Points ae epesented by a adius and an angle (, ) To plot the point 5, 4 Fist find the angle Then move out along the teminal side 5
A negative angle would be measued clockwise like usual. 3 3, 4 To plot a point with a negative adius, find the teminal side of the angle but then measue fom the pole in the negative diection of the teminal side., 3 4
Pola coodinates can also be given with the angle in degees. 10 90 60 (8, 150 135 45 30 10 ) (6, - 180 10 5 40 70 300 315 0 330 10 ) (-5, 300 ) (-3, 540 )
Let's plot the following points: 7, 7, 5 7, 3 7, Notice unlike in the ectangula coodinate system, thee ae many ways to list the same point.
Name thee othe pola coodinates that epesent that same point given: 1.) (.5, 135 ).) (-3, 60 ) (.5, -5 ) (-.5, -45 ) (-.5, -315 ) (-3, -310 ) (3, 40 ) (3, -10 )
Let's genealize the convesion fom pola to ectangula coodinates. x, y cos x x cos sin y y sin
Steps fo Conveting Equations fom Rectangula to Pola fom and vice vesa Fou citical equivalents to keep in mind ae: Actan y Actan y x x If x > 0 If x < 0
Identify, then convet to a ectangula equation and then gaph the equation: Cicle with cente at the pole and adius.
IDENTIFY and GRAPH: 3 The gaph is a staight line at the pole. extending though 3
Now convet that equation into a ectangula equation: 3 Take the tan of both sides: Actan y x 3 y x tan 3 y x tan 3 y x 3 Coss multiply: y 3x 3x y 0
IDENTIFY, GRAPH, AND THEN CONVERT TO A RECTANGULAR EQUATION: sin The gaph is a hoizontal line at y = -
GRAPH EACH POLAR EQUATION. 1.) = 3.) θ = 60
θ Gaph: cos 3 0-3 30-3.5 60-6 90 UD 10 6 150 3.5 180 3 10 3.5 40 6 70 UD 300-6 330-3.5 360-3 3 cos
IDENTIFY, GRAPH, AND THEN CONVERT TO A RECTANGULAR EQUATION: θ 0 4 30 3.5 60 90 0 10-150 -3.5 180-4 10-3.5 40-70 0 300 330 3.5 360 4 4cos
6sin
Cadioids (heat-shaped cuves) whee a > 0 and passes though the oigin a a cos a a cos 44cos 44cos
a asin a asin 4 4sin 4 4sin
54sin 53cos 54sin 54cos
45sin 35cos 5sin 45cos
4sin3 4cos3 4sin 4 4cos4
9sin Not on ou quiz! 4 5 5 4
9cos Not on ou quiz! 4 5 5 4
Can you gaph each equation on the same gaph? cos 3 3 cos Vetical Line sin3 3 sin Hoizontal Line cos 5sin 3 0 3 cos 5sin Geneal Line
Can you gaph each cicle on the same gaph?? 5 cos 4sin
Can you gaph both equations on the same gaph?? 4sin 4 cos a bsin a bcos Note: INNER loop Only if a < b
Can you gaph each spial below? 3
Gaph each equation 5cos3 6sin4 asinn n even n pedals bsin n n odd n pedals
Gaph: = + 3cosθ LIMACON WITH INNER LOOP θ 0 5 30 4.6 60 3.5 90 10 0.5 150-0.6 180-1 10-0.6 40 0.5 70 300 3.5 330 4.6 360 5
Gaph: = 3sin 3θ Rose with 3 petals θ 0 0 30 3 60 0 90-3 10 0 150 3 180 0 10-3 40 0 70 3 300 0 330-3 360 0
Let's take a point in the ectangula coodinate system and convet it to the pola coodinate system. pola coodinates ae: 3 (3, 4) 4 We'll find in adians (5, 0.93) Based on the tig you know can you see how to find and? 3 4 = 5 tan 4 3 tan 1 4 3 0.93
Let's genealize this to find fomulas fo conveting fom ectangula to pola coodinates. (x, y) x y x y x tan y x y If x > 0 y tan 1 o tan 1 y x x If x < 0
Now let's go the othe way, fom pola to ectangula coodinates. Based on the tig you know can you see how to find x and y? 4 4 cos x ectangula coodinates ae: 4, 4 4 y x 4 4 x 4 4 sin y 4 y,
Convet each of these ectangula coodinates to pola coodinates: 1.) 0, 4.) 1, 3 4, 3.) 3 4.), 1 4, 0 1, 3 1, 5 6 4, 0 1, 3 5.) 6.), 4 3,, 4
Convet each of these pola coodinates to ectangula coodinates: 1.) 6,10.) 4, 45 3, 3 3, 3.) 3, 300 4.) 0, 13 3 5.) 6.) 3, 3 3 0, 0 4, 6 3, 3 4 3, 3, 3
Steps fo Conveting Equations fom Rectangula to Pola fom and vice vesa Fou citical equivalents to keep in mind ae: Actan y Actan y x x If x > 0 If x < 0
Convet the equation: = to ectangula fom Since we know that the equation., squae both sides of
We still need, but is thee a bette choice than squaing both sides?
Convet the following equation fom ectangula to pola fom. Since x y x and x cos cos cos
Convet the following equation fom ectangula to pola fom. x y 3 (x y ) 3 3 x y 3 3
Convet the ectangula coodinate system equation to a pola coodinate system equation. x y 9 x y 3 Hee each unit is 1/ and we went out 3 and did all angles. Fom convesions, how was elated to x and y? Befoe we do the convesion let's look at the gaph. must be 3 but thee is no estiction on so conside all values.
Convet the ectangula coodinate system equation to a pola coodinate system equation. What ae the pola convesions we found fo x and y? x 4y substitute in fo x and y x cos y sin cos 4 sin cos 4 sin cos 4sin 4sin cos 4tan sec
WRITE EACH EQUATION IN POLAR FORM: 1.) y x.) x 11 5 5 cos( 153) 11 cos