CHAT Pre-Calculus Section Polar Coordinates

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1 CHAT Pe-Calculus Pola Coodinates Familia: Repesenting gaphs of equations as collections of points (, ) on the ectangula coodinate sstem, whee and epesent the diected distances fom the coodinate aes to the point (, ). New: The Pola Coodinate Sstem, which consists of a fied point O, called the pole, and a a, called the pola ais, with its initial point at O. Each point P in the plane can be labeled with pola coodinates (, θ), whee is the diected distance fom O to P and θ is an angle in standad position with teminal side at OP. diected distance P = (, θ) O θ = diected angle Pola ais 1

2 CHAT Pe-Calculus Eample: Gaph the point sstem. in the pola coodinate, , 3 0 Eample: Gaph the point sstem. 3 3, 4 in the pola coodinate 3,

3 CHAT Pe-Calculus Eample: Gaph the point sstem. 5, 4 3 in the pola coodinate 5 3, Eample: Gaph the point sstem. Remembe that is a diected distance., 4 3 in the pola coodinate 4 3,

4 CHAT Pe-Calculus Notice that the last 3 points gaphed the same point. In the pola coodinate sstem, thee ae infinitel man coodinates that gaph to the same point. This is diffeent than the ectangula coodinate sstem. In the ectangula coodinate sstem, each point has a unique epesentation. In geneal, (, ) (, n) o (, ) (, (n 1) ) whee n is an intege. Eample: Find thee othe pola epesentations fo the point, Solution:,, 7,, 3, 3, etc Coodinate Convesion To see the elationship between pola and ectangula coodinates, let the sstems coincide. 4

5 CHAT Pe-Calculus Pole (Oigin) θ (, θ) (, ) Pola ais (-ais) Because the point (, ) lies on the cicle of adius, it follows that elationships tan Coodinate Convesion. You can also see the tig cos sin The pola coodinates (, θ) ae elated to the ectangula coodinates (, ) as follows. cos sin and tan 5

6 CHAT Pe-Calculus Eample: Convet the point (3, π) to ectangula coodinates. cos 3cos 3( 1) 3 sin 3sin 3(0) 0 The ectangula coodinates ae (-3, 0). Eample: Convet the point (4, 6 ) to ectangula coodinates. cos 4cos sin 4sin The ectangula coodinates ae ( 3, ). 6

7 CHAT Pe-Calculus Eample: Convet (-1, -1) to pola coodinates. tan 1 tan 1 tan 1 Accoding to allsintancos, the tangent = 1 when θ is in Quadant I o Quadant III. Since ou point is in quadant 5 III, we will choose θ =. 4 Now find. ( 1) ( 1) Since θ lies in the same quadant as (,) we need to use the positive value of. 5 The pola coodinates ae (, ). 4 Note: If we had chosen 4 fo θ, we would have had to use fo to get the same point. 7

8 CHAT Pe-Calculus Eample: Convet (0, -) to pola coodinates. Solution: This point lies units down on the -ais. That 3 means = and θ =, giving us the 3 point (, ). Conveting Points Between Pola and Rectangula Sstems using a Gaphing Calculato Conveting Points fom Pola to Rectangula To convet the point (, ) to ectangula coodinates, do the following: 1. To find the -coodinate, pess [ nd ] [ANGLE] [PR(]. Now ente ou pola coodinates, θ and pess [ENTER].. To find the -coodinate, pess [ nd ] [ANGLE] [PR(]. Now ente ou pola coodinates, θ and pess [ENTER]. 8

9 CHAT Pe-Calculus Conveting Points fom Rectangula to Pola To convet the point (, ) to pola coodinates, do the following: 1. To find the adius, pess [ nd ] [ANGLE] [RP(]. Now ente ou ectangula coodinates, and pess [ENTER]. The value of will be in decimal fom.. To find the angle θ, pess [ nd ] [ANGLE] [RPθ (]. Now ente ou ectangula coodinates, and pess [ENTER]. If ou ae in DEGREE mode, the angle will be given in degees. If ou ae in RADIAN mode, the answe will be given in adians. *Note: The pola coodinates of a given point (, ) ae not unique. Thee ae othe possible answes. Equation Convesion To convet ectangula equations to pola equations, simpl eplace with cos and with sin. Conveting pola equations to ectangula equations equies consideable ingenuit. 9

10 CHAT Pe-Calculus Eample: Convet the ectangula equation pola equation. to a Solution: Use cos and sin and substitute. ( cos ) cos cos cos ( sin ) sin sin sin cos sin sin sin cot csc Eample: Convet the pola equation 3 to a ectangula equation. Solution: Think about what this equation looks like. It is all the points that ae 3 units fom the pole. This is a cicle of adius 3. In ectangula fom, this equation would be 9. 10

11 CHAT Pe-Calculus Eample: Convet the pola equation ectangula equation. to a 6 Solution: Think about what this equation looks like. It is all of the points that lie on the line that makes an angle of with the positive pola ais. 6 We know that tan, so we get tan tan This answe fits what we would epect. We should get a linea equation of the fom = m, since this line goes though the oigin with positive slope. 11

12 Eample: Convet the pola equation ectangula equation. sec to a CHAT Pe-Calculus Solution: What we have to wok with is cos sin and tan Since secant and cosine ae elated, we stat with sec. sec 1 cos cos 1 We know that cos, we must have the equation 1 This is ou ectangula equation. It is a vetical line at 1. 1

13 CHAT Pe-Calculus Eample: Convet the pola equation ectangula equation. 1 to a 1 sin Solution: Clea the faction and wok with the equation until something useful o familia comes up. 1 1 sin (1 sin ) 1 sin 1 We know that sin sin cos so substitute. sin 1 (sin cos ) 1 (sin cos ) 1 Thee is an cos and sin embedded hee. ( sin )( cos ) 1 Substitute fo cos, sin, and to get 1 13

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