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Read Symbol defintion
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Q01 |
- Solve x^5 + 1 = 0
- Q02 | - Solve x^5 - 1 = 0
- Q03 | - Solve x^5 + i = 0
- Q04 | - Solve x^5 - i = 0
- Q05 | - Show that 0.30901 + 0.95106*i is a root of x^5 - 1 = 0
- Q06 | -
- Q07 | -
- Q08 | -
- Q09 | -
- Q10 | -
- Q02 | - Solve x^5 - 1 = 0
Q01. Solve x^5 + 1 = 0
De Moivre's Theory : Draw a unit circle
- 1. Draw 1st point on circle with angle : A1 = 036
- 2. Draw 2nd point on circle with angle : A2 = 108
- 3. Draw 3nd point on circle with angle : A3 = 180
- 4. Draw 4nd point on circle with angle : A3 = 252
- 5. Draw 5nd point on circle with angle : A3 = 324
- r1 = cos(A1) + i*sin(A1)
- r2 = cos(A2) + i*sin(A2)
- r3 = cos(A3) + i*sin(A3)
- r4 = cos(A4) + i*sin(A4)
- r5 = cos(A5) + i*sin(A5)
- For angle A1 : r1 = x1 + i*y1. Measure x1 and y1
- For angle A2 : r2 = x2 + i*y2. Measure x2 and y2
- Similarly find other five solutions
- Use the conjugate complex to reduce the calculation
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Q02. Solve x^5 - 1 = 0
De Moivre's Theory : Draw a unit circle
- 1. Draw 1st point on circle with angle : A1 = 000
- 2. Draw 2nd point on circle with angle : A2 = 072
- 3. Draw 3nd point on circle with angle : A3 = 144
- 4. Draw 4nd point on circle with angle : A3 = 216
- 5. Draw 5nd point on circle with angle : A3 = 288
- r1 = cos(A1) + i*sin(A1)
- r2 = cos(A2) + i*sin(A2)
- r3 = cos(A3) + i*sin(A3)
- r4 = cos(A4) + i*sin(A4)
- r5 = cos(A5) + i*sin(A5)
- For angle A1 : r1 = x1 + i*y1. Measure x1 and y1
- For angle A2 : r2 = x2 + i*y2. Measure x2 and y2
- Similarly find other five solutions
- Use the conjugate complex to reduce the calculation
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Q03. Solve x^5 + i = 0
De Moivre's Theory : Draw a unit circle
- 1. Draw 1st point on circle with angle : A1 = 054
- 2. Draw 2nd point on circle with angle : A2 = 126
- 3. Draw 3nd point on circle with angle : A3 = 198
- 4. Draw 4nd point on circle with angle : A3 = 270
- 5. Draw 5nd point on circle with angle : A3 = 342
- r1 = cos(A1) + i*sin(A1)
- r2 = cos(A2) + i*sin(A2)
- r3 = cos(A3) + i*sin(A3)
- r4 = cos(A4) + i*sin(A4)
- r5 = cos(A5) + i*sin(A5)
- For angle A1 : r1 = x1 + i*y1. Measure x1 and y1
- For angle A2 : r2 = x2 + i*y2. Measure x2 and y2
- Similarly find other five solutions
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Q04. x^5 - i = 0
De Moivre's Theory : Draw a unit circle
- 1. Draw 1st point on circle with angle : A1 = 018
- 2. Draw 2nd point on circle with angle : A2 = 090
- 3. Draw 3nd point on circle with angle : A3 = 162
- 4. Draw 4nd point on circle with angle : A3 = 234
- 5. Draw 5nd point on circle with angle : A3 = 306
- r1 = cos(A1) + i*sin(A1)
- r2 = cos(A2) + i*sin(A2)
- r3 = cos(A3) + i*sin(A3)
- r4 = cos(A4) + i*sin(A4)
- r5 = cos(A5) + i*sin(A5)
- For angle A1 : r1 = x1 + i*y1. Measure x1 and y1
- For angle A2 : r2 = x2 + i*y2. Measure x2 and y2
- Similarly find other five solutions
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Q05. Show that 0.30901 + 0.95106*i is a root of x^5 - 1 = 0
Proof
- x^5 - 1 = (0.30901 + 0.95106*i)^5 - 1
- = (0.30901)^5
- + 05*((0.30901)^4)*((0.95106)^1)*i
- + 10*((0.30901)^3)*((0.95106)^2)*(i^2)
- + 10*((0.30901)^2)*((0.95106)^3)*(i^3)
- + 05*((0.30901)^1)*((0.95106)^4)*(i^4)
- + 01**((0.95106)^5)*(i^5)
- - 1
- = 0.002817 + 5*0.0091178*0.95106*i + 10*0.0295064*0.904515*(-1)
- + 10*0.09548718*0.860248*(-i) + 5*0.30901*0.8181476*(+1) + 0.778107*(i) - 1
- = 0.02817 + 0.0433578*i - 0.2668898 - 0.821426556*i + 1.2640789 + 0.778107*i - 1
- = 0.0000 + 0.0000*i
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Q06.
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Q07.
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Q08.
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Q09. Answer
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Q10. Answer
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