chapter 40 cartesian and polar co-ordinates - AWS
CHAPTER 40 CARTESIAN AND POLAR COORDINATES EXERCISE 169 Page 462
1. Express (3, 5) as polar coordinates, correct to 2 decimal places, in both degrees and in radians.
From the diagram,= r
32 + 52 = 5.83
5 y π −1 and= rad = 1.03 rad tan −1 = 59.04° or 59.04 × θ tan = 180 3 x
Hence, (3, 5) in Cartesian coordinates corresponds to (5.83, 59.04°) or (5.83, 1.03 rad) in polar coordinates
2. Express (6.18, 2.35) as polar coordinates, correct to 2 decimal places, in both degrees and in radians.
From the diagram, = r
6.182 + 2.352 = 6.61
y 2.35 π −1 and= = 20.82° or 20.82 × rad = 0.36 rad = θ tan tan −1 x 6.18 180
Hence, (6.18, 2.35) in Cartesian coordinates corresponds to (6.61, 20.82°) or (6.61, 0.36 rad) in polar coordinates
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© 2014, John Bird
3. Express (–2, 4) as polar coordinates, correct to 2 decimal places, in both degrees and in radians.
From the diagram,= r
22 + 42 = 4.47
y 4 −1 = α tan = tan −1 = 63.43° x 2
and
θ = 180° – α = 116.57° or 116.57 ×
Thus,
π 180
rad = 2.03 rad
Hence, (–2, 4) in Cartesian coordinates corresponds to (4.47, 116.57°) or (4.47, 2.03 rad) in polar coordinates 4. Express (–5.4, 3.7) as polar coordinates, correct to 2 decimal places, in both degrees and in radians.
From the diagram, r =
5.42 + 3.7 2 = 6.55
y 3.7 −1 = 34.42° tan −1 α tan = = x 5.4
and Thus,
θ = 180° – α = 145.58° or 145.58 ×
π 180
rad = 2.54 rad
Hence, (–5.4, 3.7) in Cartesian coordinates corresponds to (6.55, 145.58°) or (6.55, 2.54 rad) in polar coordinates
5. Express (–7, –3) as polar coordinates, correct to 2 decimal places, in both degrees and in radians. 680
© 2014, John Bird
From the diagram,= r
7 2 + 32 = 7.62 3 = 23.20° 7
and
α = tan −1
Thus,
θ = 180° + 23.20° = 203.20° or 203.20 ×
π 180
rad = 3.55 rad
Hence, (–7, –3) in Cartesian coordinates corresponds to (7.62, 203.20°) or (7.62, 3.55 rad) in polar coordinates
6. Express (–2.4, –3.6) as polar coordinates, correct to 2 decimal places, in both degrees and in radians.
From the diagram, r =
2.42 + 3.62 = 4.33 3.6 = 56.31° 2.4
and
α = tan −1
Thus,
θ = 180° + 56.31° = 236.31° or 236.31×
π 180
rad = 4.12 rad
Hence, (–2.4, –3.6) in Cartesian coordinates corresponds to (4.33, 236.31°) or (4.33, 4.12 rad) in polar coordinates
7. Express (5, –3) as polar coordinates, correct to 2 decimal places, in both degrees and in radians.
From the diagram,= r
52 + 32 = 5.83 681
© 2014, John Bird
3 = 30.96° 5
and
α = tan −1
Thus,
θ = 360° – 30.96° = 329.04° or 329.04 ×
π 180
rad = 5.74 rad
Hence, (5, –3) in Cartesian coordinates corresponds to (5.83, 329.04°) 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 radians.
From the diagram, = r
9.62 + 12.42 = 15.68 12.4 = 52.25° 9.6
and
α = tan −1
Thus,
θ = 360° – 52.25° = 307.75° or 307.75 ×
π 180
rad = 5.37 rad
Hence, (9.6, –12.4) in Cartesian coordinates corresponds to (15.68, 307.75°) or (15.68, 5.37 rad) in polar coordinates
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© 2014, John Bird
EXERCISE 170 Page 463
1. Express (5, 75°) as Cartesian coordinates, correct to 3 decimal places.
In the diagram,
x = 5 cos 75° = 1.294
and
y = 5 sin 75° = 4.830
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 = 1.917
and
y = 4.4 sin 1.12 = 3.960
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° = –5.362
and
y = 7 sin 140° = 4.500
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 = –2.884 683
© 2014, John Bird
y = 3.6 sin 2.5 rad = 2.154 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° = –9.353 y = 10.8 sin 210° = –5.400 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 = –2.615 y = 4 sin 4 rad = –3.027 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° = 0.750 y = 1.5 sin 300° = –1.299 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 = 4.252 y = 6 sin 5.5 rad = –4.233 Hence, (6, 5.5 rad) in polar form corresponds to (4.252, – 4.233) in Cartesian form
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© 2014, John Bird
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 40∠90°. The holes will be equally displaced,
360° i.e. 72° apart. 5
Thus, in polar form the holes are at (40, 90°), (40, (90° + 72°)), i.e. (40, 162°), (40, (162° + 72°)), i.e. (40, 234°), (40, (234° + 72°)), i.e. (40, 306°), and (40, (306° + 72°)), 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
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© 2014, John Bird
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 = 40 – 12.36 = 27.64, and BC = 38.04 Thus, by Pythagoras’ theorem, AB =
( 27.642 + 38.042 )
= 47.0 mm
i.e. the shortest distance between the centres of two adjacent holes is 47.0 mm
686
© 2014, John Bird
1. Express (3, 5) as polar coordinates, correct to 2 decimal places, in both degrees and in radians.
From the diagram,= r
32 + 52 = 5.83
5 y π −1 and= rad = 1.03 rad tan −1 = 59.04° or 59.04 × θ tan = 180 3 x
Hence, (3, 5) in Cartesian coordinates corresponds to (5.83, 59.04°) or (5.83, 1.03 rad) in polar coordinates
2. Express (6.18, 2.35) as polar coordinates, correct to 2 decimal places, in both degrees and in radians.
From the diagram, = r
6.182 + 2.352 = 6.61
y 2.35 π −1 and= = 20.82° or 20.82 × rad = 0.36 rad = θ tan tan −1 x 6.18 180
Hence, (6.18, 2.35) in Cartesian coordinates corresponds to (6.61, 20.82°) or (6.61, 0.36 rad) in polar coordinates
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© 2014, John Bird
3. Express (–2, 4) as polar coordinates, correct to 2 decimal places, in both degrees and in radians.
From the diagram,= r
22 + 42 = 4.47
y 4 −1 = α tan = tan −1 = 63.43° x 2
and
θ = 180° – α = 116.57° or 116.57 ×
Thus,
π 180
rad = 2.03 rad
Hence, (–2, 4) in Cartesian coordinates corresponds to (4.47, 116.57°) or (4.47, 2.03 rad) in polar coordinates 4. Express (–5.4, 3.7) as polar coordinates, correct to 2 decimal places, in both degrees and in radians.
From the diagram, r =
5.42 + 3.7 2 = 6.55
y 3.7 −1 = 34.42° tan −1 α tan = = x 5.4
and Thus,
θ = 180° – α = 145.58° or 145.58 ×
π 180
rad = 2.54 rad
Hence, (–5.4, 3.7) in Cartesian coordinates corresponds to (6.55, 145.58°) or (6.55, 2.54 rad) in polar coordinates
5. Express (–7, –3) as polar coordinates, correct to 2 decimal places, in both degrees and in radians. 680
© 2014, John Bird
From the diagram,= r
7 2 + 32 = 7.62 3 = 23.20° 7
and
α = tan −1
Thus,
θ = 180° + 23.20° = 203.20° or 203.20 ×
π 180
rad = 3.55 rad
Hence, (–7, –3) in Cartesian coordinates corresponds to (7.62, 203.20°) or (7.62, 3.55 rad) in polar coordinates
6. Express (–2.4, –3.6) as polar coordinates, correct to 2 decimal places, in both degrees and in radians.
From the diagram, r =
2.42 + 3.62 = 4.33 3.6 = 56.31° 2.4
and
α = tan −1
Thus,
θ = 180° + 56.31° = 236.31° or 236.31×
π 180
rad = 4.12 rad
Hence, (–2.4, –3.6) in Cartesian coordinates corresponds to (4.33, 236.31°) or (4.33, 4.12 rad) in polar coordinates
7. Express (5, –3) as polar coordinates, correct to 2 decimal places, in both degrees and in radians.
From the diagram,= r
52 + 32 = 5.83 681
© 2014, John Bird
3 = 30.96° 5
and
α = tan −1
Thus,
θ = 360° – 30.96° = 329.04° or 329.04 ×
π 180
rad = 5.74 rad
Hence, (5, –3) in Cartesian coordinates corresponds to (5.83, 329.04°) 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 radians.
From the diagram, = r
9.62 + 12.42 = 15.68 12.4 = 52.25° 9.6
and
α = tan −1
Thus,
θ = 360° – 52.25° = 307.75° or 307.75 ×
π 180
rad = 5.37 rad
Hence, (9.6, –12.4) in Cartesian coordinates corresponds to (15.68, 307.75°) or (15.68, 5.37 rad) in polar coordinates
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© 2014, John Bird
EXERCISE 170 Page 463
1. Express (5, 75°) as Cartesian coordinates, correct to 3 decimal places.
In the diagram,
x = 5 cos 75° = 1.294
and
y = 5 sin 75° = 4.830
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 = 1.917
and
y = 4.4 sin 1.12 = 3.960
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° = –5.362
and
y = 7 sin 140° = 4.500
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 = –2.884 683
© 2014, John Bird
y = 3.6 sin 2.5 rad = 2.154 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° = –9.353 y = 10.8 sin 210° = –5.400 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 = –2.615 y = 4 sin 4 rad = –3.027 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° = 0.750 y = 1.5 sin 300° = –1.299 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 = 4.252 y = 6 sin 5.5 rad = –4.233 Hence, (6, 5.5 rad) in polar form corresponds to (4.252, – 4.233) in Cartesian form
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© 2014, John Bird
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 40∠90°. The holes will be equally displaced,
360° i.e. 72° apart. 5
Thus, in polar form the holes are at (40, 90°), (40, (90° + 72°)), i.e. (40, 162°), (40, (162° + 72°)), i.e. (40, 234°), (40, (234° + 72°)), i.e. (40, 306°), and (40, (306° + 72°)), 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
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© 2014, John Bird
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 = 40 – 12.36 = 27.64, and BC = 38.04 Thus, by Pythagoras’ theorem, AB =
( 27.642 + 38.042 )
= 47.0 mm
i.e. the shortest distance between the centres of two adjacent holes is 47.0 mm
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© 2014, John Bird
