Polar Coordinates. a) (2; 30 ) b) (5; 120 ) c) (6; 270 ) d) (9; 330 ) e) (4; 45 )

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1 Pola Coodinates We now intoduce anothe method of labelling oints in a lane. We stat by xing a oint in the lane. It is called the ole. A standad choice fo the ole is the oigin (0; 0) fo the Catezian coodinate system.. Pole Pola axis We then x a ay stating fom the ole. It is called the ola axis. A standad choice fo the ola axis is the ositive half of the hoizontal axis. To detemine the ola coodinates of a given oint P in the lane, do the following: (i) Detemine its distance fom the ole. (ii) Detemine the angle between the ola axis and the ay OP. Then the ola coodinates of P ae (; ). In the gue below, O is the ole, and OA is the ola axis. The given oint P is units fom the ole and the ay OP makes an angle of 0 with the ola axis. Theefoe its ola coodinates ae (; 0 ). Thee ae secial ola gah aes that one may easily use to lot oints in ola coodinates. The gue below is a simle examle. It consists of cicles with the same cente O and di eent adii. The common cente O is the ole. The ay labelled 0 is the ola axis. Rays making angles of 0, 60 ; : : : ae also dawn. Conside the oint P maked by *. It is 8 units fom the ole, because it is on a cicle centeed at the ole with adius 8, and the ay OP makes an angle of 0 with the ola axis. Theefoe P has ola coodinates (8; 0 ). Plot the following oints on the above ola gah ae. a) (; 0 ) b) (5; 10 ) c) (6; 70 ) d) (9; 0 ) e) (4; 45 ) f) (7:5; 110 ) g) (8:; 75 ) h) (9:5; 40 ) i) (:6; 0 ) j) (1:8; 00 ) 1

2 Conveting Catesian coodinates into ola coodinates Fo an examle, suose we ae equied to detemine the ola coodinates of the oint with Catesian coodinates q ( 4; 4). It is lotted in the gue below and it is denoted by P. Its distance fom the oigin (0; 0) is ( 4) + 4 = 4. The angle between the ay OP and the ositive hoizontal axis should be easy to guess because the ay bisects the ight angle between the negative hoizontal axis and the ositive vetical axis. It is 15 and so the ola coodinates fo ( 4; 4) ae 4 ; P * O 4 6 In geneal, to detemine the ola coodinates of a oint with Catesian coodinates (x; y), you have to: Detemine the distance fom (0; 0) to (x; y). It is given by the fomula = x + y : Detemine the angle between the ositive hoizontal axis and the ay fom the oigin (0; 0) to the oint (x; y). If the angle is then tan = y x. Detemine its value fom the fact that it is in the same quadant as the oint (x; y). Examle 1 To detemine the ola coodinates of the oint with Catesian coodinates 1; : Solution The oint is lotted in the gue below. Its distance fom the oigin is given by = 1 + = 1 + = 4 =

3 * The angle between the ositive hoizontal axis and the ay fom (0; 0) to 1; condition tan = 1. Since it is in the thid quadant, = ( ) = 40 satis es the Theefoe the ola coodinates fo 1; ae (; 40 ) Execise 1. Plot the oints with the given ola coodinates on the given ola gah a) (5; 60 ) b) (7:5; 40 ) c) (9; 10 ) d) (10; 00 ). The ola coodinates fo the oint with Catesian coodinates (5; 5) ae (A) (5; 45 ) (B) 5 5; 45 (C) 5 ; 45 (D) 5 ; 60. What ae the ola coodinates of the oint with Catesian coodinates (0; 9)? Pola Fom Of A Comlex Numbe The st ste in detemining the so-called ola fom of a given comlex numbe x + yi is to eesent it by a oint in a lane. So, we daw the usual hoizontal and vetical Catesian coodinates lines as shown below then eesented the comlex numbe by the oint (x; y). Examle Conside the comlex numbes + i, 4i, 4:5 i and 4 + 4i. + i is eesented by the oint * with Catesian coodinates ( ; ). 4i is eesented by the oint with Catesian coodinates ( ; 4). 4:5 i is eesented by the oint # with Catesian coodinates (4:5; ) i is eesented by the oint with Catesian coodinates (4; 4).

4 The next ste is to convet the Catesian coodinates (x; y) of the oint into ola coodinates. Thus we detemine = a + b and = tan 1 y x. Examle 4 In the case of 4 + 4i which was eesented by the oint with Catesian coodinates (4; 4), we get = = 4 and = tan 1 1 = 4 o 45, if we choose to measue angles in degees, and so the ola coodinates of (4; 4) ae 4 ; 4 o 4 ; 45. Now note that the eal at of 4 + 4i is cos 4 = 4 cos 4 and the comlex at is sin 4 = 4 sin 4. Theefoe 4 + 4i = 4 cos i sin 4 = 4 (cos 4 + i sin 4 ) This is the ola fom of 4 + 4i. Going back to an abitay comlex numbe x + yi, if the ola coodinates fo (x; y) ae (; ), whee = x + y and = tan 1 y x, then Theefoe x = cos and y = sin x + yi = cos + i sin = (cos + i sin ) This is called the ola fom of the comlex numbe x + yi. The distance is called the absolute value of x + yi and the angle is called its agument. Examle 5 To detemine the ola fom of the comlex numbe 5i: The numbe is lotted below. Its absolute value is = + 5 = 4. Its agument is an angle u in the d quadant that satis es the equation tan u = 5 = 5 5. Thus the efeence angle fo u is tan 1 = 59, to the neaest degee. It follows that u = ( ) = 9. The equied ola fom is 5i = 4 (cos 9 + i sin 9 ) Multilying/Dividing comlex numbes in ola fom Conside comlex numbes z = (cos u + i sin u) and w = l (cos v + i sin v) in ola fom. Thei oduct zw is zw = (cos u + i sin u) l (cos v + i sin v) = l (cos u + i sin u) (cos v + i sin v) = l [(cos u cos v sin u sin v) + i (sin u cos v + cos u sin v)] = l [cos(u + v) + i sin(u + v)] Note that the absolute value of the oduct is l which is the oduct of the absolute values of z and w. The agument of zw IS NOT the oduct of the two aguments; it is u + v, which is the sum of the two agument. Conclusion: To multily two comlex numbes in ola fom, simly multily thei absolute values, then add thei aguments. Examle: The oduct of z = 4 (cos 4 + i sin 4 ) and w = :1 (cos i sin 165 ) is the comlex numbe zw = (4) (:1) cos ( ) + i sin ( ) = 1:4 (cos 07 + i sin 07 ) 4

5 The quotient z w is z w (cos u + i sin u) (cos u + i sin u) (cos u + i sin u) (cos v i sin v) = l (cos v + i sin v) = l (cos v + i sin v) = l (cos v + i sin v) (cos v i sin v) = (cos u + i sin u) (cos v i sin v) [(cos u cos v + sin u sin v) + i (sin u cos v l (cos v + i sin v) (cos v i sin v) = l cos v + sin v = [cos(u v) + i sin(u v)] l cos u sin v)] Thus the absolute value of z w of is l and its agument is u v. Conclusion: To divide a comlex numbe z by a comlex numbe w, both in ola fom, divide the absolute value of z by the absolute value of w, then subtact the agument of w fom the agument of z. Examle: Examle: Let z = 18 (cos 1 + i sin 1 ) and w = :4 (cos 65 + i sin 65 ). Then z w = 18 cos (1 65) + i sin (1 65) = 15 :4 (cos 58 + i sin 58 ) Let z = :1 (cos 60 + i sin 60 ) and w = :4 (cos i sin 105 ). Then z w = :1 cos (60 105) + i sin (60 105) = 7 :4 8 (cos ( 45 ) + i sin ( 45 )) = 7 8 (cos 45 i sin 45 ) = i Examle 6 To nd the squae oots of z = squae is 4i 4i. In othe wods, we must nd a comlex numbe whose The st ste is to wite 4i in ola fom: Its absolute value is 4 and its agument is 70. Theefoe 4i = 4 (cos 70 + i sin 70 ). Let its squae oot be w = (cos u + i sin u). In othe wods, let w be a numbe that satis es w = 4i. We note that Theefoe w = ww = (cos u + i sin u) (cos u + i sin u) = 4 (cos 70 + i sin 70 ) It follows that = 4 and u = 70. Theefoe = and u = 15. Conclusion: A squae oot of 4i is (cos 15 + i sin 15 1 ) = + i = + i. (Squae it and see what you get.) To get a second squae oot, wite 4i as 4 cos ( ) + i sin ( ) = 4 (cos 60 + i sin 60 ). Now (cos u + i sin u) = 4 (cos 60 + i sin 60 ) Following the above foot-stes, we get = 4 and u = 60. This gives = and u = 15. Theefoe anothe squae oot of 4i is (cos 15 + i sin 15 1 i ) = = i. Execise 7 1. Plot the following oints on the ola gah ae below. a) (6; 75 ) b) (9:5; 70 ) c) (9; 0 ) d) (5; 05 ). You ae given the comlex numbes z 1 = 4 4i, z = i, z = 1 i, z4 = 5 + 5i. 5

6 (a) Daw axes and lot each numbe. 4 4i i 1 i (b) Detemine the ola fom of the numbes z 1, z, z and z 4 above. (i) Absolute value of z is Agument of z is Pola fom of z is (ii) Absolute value of z is Agument of z is Pola fom of z is (iii) Absolute value of z 4 is Agument of z 4 is Pola fom of z 4 is 5 + 5i 6

7 (c) Detemine z 1 z and leave you answe in ola fom z 1 z = (d) Detemine z z z z = and leave you answe in ola fom. (e) Detemine z 1 z 4 and leave you answe in ola fom z 1 z 4 =. What is the ola fom of the comlex numbe z = i? (A) (cos 10 + i sin 10 ) (B) 4 (cos 60 + i sin 60 ) (C) (cos 00 + i sin 00 ) (D) 4 (cos 00 + i sin 00 ) 4. You ae given the comlex numbe z = + i. (a) Detemine the ola fom of z. (b) Find the two squae oots of z. You may give you answes in ola fom. 7