CHAPTER 40 CARTESIAN AND POLAR COORDINATES
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1 CHAPTER 40 CARTESIAN AND POLAR COORDINATES EXERCISE 169 Page Express (3, 5) as polar coordinates, correct to 2 decimal places, in both degrees and in From the diagram, r = = 5.83 y and 5 θ = tan 1 = tan 1 = or rad = 1.03 rad x 3 Hence, (3, 5) in Cartesian coordinates corresponds to (5.83, ) or (5.83, 1.03 rad) 2. Express (6.18, 2.35) as polar coordinates, correct to 2 decimal places, in both degrees and in From the diagram, r = = 6.61 and tan y θ = tan 1 x = = or rad = 0.36 rad 6.18 Hence, (6.18, 2.35) in Cartesian coordinates corresponds to (6.61, ) or (6.61, 0.36 rad) 679
2 3. Express ( 2, 4) as polar coordinates, correct to 2 decimal places, in both degrees and in From the diagram, r = = 4.47 and tan y α = 4 1 = tan 1 = x 2 Thus, θ = α = or rad = 2.03 rad Hence, ( 2, 4) in Cartesian coordinates corresponds to (4.47, ) or (4.47, 2.03 rad) 4. Express ( 5.4, 3.7) as polar coordinates, correct to 2 decimal places, in both degrees and in From the diagram, r = = 6.55 and tan y α = = tan 1 = x 5.4 Thus, θ = α = or rad = 2.54 rad Hence, ( 5.4, 3.7) in Cartesian coordinates corresponds to (6.55, ) or (6.55, 2.54 rad) 5. Express ( 7, 3) as polar coordinates, correct to 2 decimal places, in both degrees and in 680
3 From the diagram, r = = 7.62 and α = tan 3 1 = Thus, θ = = or rad = 3.55 rad Hence, ( 7, 3) in Cartesian coordinates corresponds to (7.62, ) or (7.62, 3.55 rad) 6. Express ( 2.4, 3.6) as polar coordinates, correct to 2 decimal places, in both degrees and in From the diagram, r = = 4.33 and α = tan = Thus, θ = = or rad = 4.12 rad Hence, ( 2.4, 3.6) in Cartesian coordinates corresponds to (4.33, ) or (4.33, 4.12 rad) 7. Express (5, 3) as polar coordinates, correct to 2 decimal places, in both degrees and in From the diagram, r = =
4 and α = tan 3 1 = Thus, θ = = or rad = 5.74 rad Hence, (5, 3) in Cartesian coordinates corresponds to (5.83, ) or (5.83, 5.74 rad) in polar coordinates 8. Express (9.6, 12.4) as polar coordinates, correct to 2 decimal places, in both degrees and in From the diagram, r = = and α = tan = Thus, θ = = or rad = 5.37 rad Hence, (9.6, 12.4) in Cartesian coordinates corresponds to (15.68, ) or (15.68, 5.37 rad) 682
5 EXERCISE 170 Page Express (5, 75 ) as Cartesian coordinates, correct to 3 decimal places. In the diagram, x = 5 cos 75 = and y = 5 sin 75 = Hence, (5, 75 ) in polar form corresponds to (1.294, 4.830) in Cartesian form 2. Express (4.4, 1.12 rad) as Cartesian coordinates, correct to 3 decimal places. In the diagram, x = 4.4 cos 1.12 = and y = 4.4 sin 1.12 = Hence, (4.4, 1.12 rad) in polar form corresponds to (1.917, 3.960) in Cartesian form 3. Express (7,140 ) as Cartesian coordinates, correct to 3 decimal places. In the diagram, x = 7 cos 140 = and y = 7 sin 140 = Hence, (7,140 ) in polar form corresponds to ( 5.362, 4.500) in Cartesian form 4. Express (3.6, 2.5 rad) as Cartesian coordinates, correct to 3 decimal places. x = 3.6 cos 2.5 rad =
6 y = 3.6 sin 2.5 rad = Hence, (3.6, 2.5 rad) in polar form corresponds to ( 2.884, 2.154) in Cartesian form 5. Express (10.8, 210 ) as Cartesian coordinates, correct to 3 decimal places. x = 10.8 cos 210 = y = 10.8 sin 210 = Hence, (10.8, 210 ) in polar form corresponds to ( 9.353, 5.400) in Cartesian form 6. Express (4, 4 rad) as Cartesian coordinates, correct to 3 decimal places x = 4 cos rad = y = 4 sin 4 rad = Hence, (4, 4 rad) in polar form corresponds to ( 2.615, 3.027) in Cartesian form 7. Express (1.5, 300 ) as Cartesian coordinates, correct to 3 decimal places x = 1.5 cos 300 = y = 1.5 sin 300 = Hence, (1.5, 300 ) in polar form corresponds to (0.750, 1.299) in Cartesian form 8. Express (6, 5.5 rad) as Cartesian coordinates, correct to 3 decimal places. x = 6 cos 5.5 rad = y = 6 sin 5.5 rad = Hence, (6, 5.5 rad) in polar form corresponds to (4.252, 4.233) in Cartesian form 684
7 9. The diagram below shows five equally spaced holes on an 80 mm pitch circle diameter. Calculate their coordinates relative to axes 0x and 0y in (a) polar form, (b) Cartesian form. (a) In the diagram below, hole A is at an angle of 90. Hence, in polar form, hole A is The holes will be equally displaced, 360 i.e. 72 apart. 5 Thus, in polar form the holes are at (40, 90 ), (40, ( )), i.e. (40, 162 ), (40, ( )), i.e. (40, 234 ), (40, ( )), i.e. (40, 306 ), and (40, ( )), i.e. (40, 378 ) or (40, 18 ). Summarizing, the holes are at (40, 18 ), (40, 90 ), (40, 162 ), (40, 234 ) and (40, 306 ) (b) (40, 18 ) = (40 cos 18, 40 sin 18 ) = (38.04, 12.36) in Cartesian form (40, 90 ) = (40 cos 90, 40 sin 90 ) = (0, 40) in Cartesian form (40, 162 ) = (40 cos 162, 40 sin 162 ) = ( 38.04, 12.36) in Cartesian form (40, 234 ) = (40 cos 234, 40 sin 234 ) = ( 23.51, 32.36) in Cartesian form (40, 306 ) = (40 cos 306, 40 sin 306 ) = (23.51, 32.36) in Cartesian form 685
8 10. In the diagram of Problem 9, calculate the shortest distance between the centres of two adjacent holes. In triangle ABC in the above diagram, AC = = 27.64, and BC = Thus, by Pythagoras theorem, AB = ( ) + = 47.0 mm i.e. the shortest distance between the centres of two adjacent holes is 47.0 mm 686
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