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Remembering that any complex number can be written in the form re" by (9.4), we get Section 9 Euler's Formula 63 2 e2 = ri r2 e4(02+@2), e 21 22 (9.6) In words, to multiply two complex numbers, we multiply their absolute values and add their angles. To divide two complex numbers, we divide the absolute values and subtract the angles Example. Evaluate (1 + i)?/(1 - i). From Figure 5.1 we have 1 2et/4. We plot 1 - i in Figure 9.5 and find r2, 0=/4 (or +7/4), so 1-i= /2e-i#/4. Then (VEetr/a)2 V2e-i7/4=J%-in/s = /2e3i#/4 2 eir/2 (1+i)2 1-1 Figure 9.5 From Figure 9.6, we find =-1, y = 1, so (1i)2 1-1 =riy-1+i We could use degrees in this problem. By (9.6), we find that the angle of (1 i)2/(1-i) is 2(45°) - (-45°) 135° as in Figure 9.6 Figure 9.6 PROBLEMS, SECTION 9 Express the following complex numbers in the r + iy form. Try to visualize each complex number, using sketches as in the examples if necessary. The first twelve problems you should be able to do in your head (and maybe some of the others-try it!) Doing a problem quickly in your head saves time over using a computer. Remember that the point in doing problems like this is to gain skill in manipulating complex expressions, so a good study method is to do the problems by hand and use a computer to check your answers. 3. 93ri/2 e-2i -4mi - 2. ei/2 1. ei/4 (a/3)(344mi 4. 6. 5. 7. 3e2(1+i 2esri/6 9. 2e-i/2 /4 4e-Sin/3 10. 11. 12. (i) 1-i (1+ W) 15. (1 (1 i)* 14. 13. ( (-)(1+v) 17. 16 () 19. (1-) 21 20.

Remembering that any complex number can be written in the form re" by (9.4), we get Section 9 Euler's Formula 63 2 e2 = ri r2 e4(02+@2), e 21 22 (9.6) In words, to multiply two complex numbers, we multiply their absolute values and add their angles. To divide two complex numbers, we divide the absolute values and subtract the angles Example. Evaluate (1 + i)?/(1 - i). From Figure 5.1 we have 1 2et/4. We plot 1 - i in Figure 9.5 and find r2, 0=/4 (or +7/4), so 1-i= /2e-i#/4. Then (VEetr/a)2 V2e-i7/4=J%-in/s = /2e3i#/4 2 eir/2 (1+i)2 1-1 Figure 9.5 From Figure 9.6, we find =-1, y = 1, so (1i)2 1-1 =riy-1+i We could use degrees in this problem. By (9.6), we find that the angle of (1 i)2/(1-i) is 2(45°) - (-45°) 135° as in Figure 9.6 Figure 9.6 PROBLEMS, SECTION 9 Express the following complex numbers in the r + iy form. Try to visualize each complex number, using sketches as in the examples if necessary. The first twelve problems you should be able to do in your head (and maybe some of the others-try it!) Doing a problem quickly in your head saves time over using a computer. Remember that the point in doing problems like this is to gain skill in manipulating complex expressions, so a good study method is to do the problems by hand and use a computer to check your answers. 3. 93ri/2 e-2i -4mi - 2. ei/2 1. ei/4 (a/3)(344mi 4. 6. 5. 7. 3e2(1+i 2esri/6 9. 2e-i/2 /4 4e-Sin/3 10. 11. 12. (i) 1-i (1+ W) 15. (1 (1 i)* 14. 13. ( (-)(1+v) 17. 16 () 19. (1-) 21 20.

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
8th Edition
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter8: Complex Numbers And Polarcoordinates
Section8.4: Roots Of A Complex Number
Problem 41PS: Recall from the introduction to Section 8.2 that Jerome Cardans solutions to the equation x3=15x+4...
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Problems, Section 9 number 8,10,11,13. Please answer the questions

Remembering that any complex number can be written in the form re" by (9.4), we get Section 9 Euler's Formula 63 2 e2 = ri r2 e4(02+@2), e 21 22 (9.6) In words, to multiply two complex numbers, we multiply their absolute values and add their angles. To divide two complex numbers, we divide the absolute values and subtract the angles Example. Evaluate (1 + i)?/(1 - i). From Figure 5.1 we have 1 2et/4. We plot 1 - i in Figure 9.5 and find r2, 0=/4 (or +7/4), so 1-i= /2e-i#/4. Then (VEetr/a)2 V2e-i7/4=J%-in/s = /2e3i#/4 2 eir/2 (1+i)2 1-1 Figure 9.5 From Figure 9.6, we find =-1, y = 1, so (1i)2 1-1 =riy-1+i We could use degrees in this problem. By (9.6), we find that the angle of (1 i)2/(1-i) is 2(45°) - (-45°) 135° as in Figure 9.6 Figure 9.6 PROBLEMS, SECTION 9 Express the following complex numbers in the r + iy form. Try to visualize each complex number, using sketches as in the examples if necessary. The first twelve problems you should be able to do in your head (and maybe some of the others-try it!) Doing a problem quickly in your head saves time over using a computer. Remember that the point in doing problems like this is to gain skill in manipulating complex expressions, so a good study method is to do the problems by hand and use a computer to check your answers. 3. 93ri/2 e-2i -4mi - 2. ei/2 1. ei/4 (a/3)(344mi 4. 6. 5. 7. 3e2(1+i 2esri/6 9. 2e-i/2 /4 4e-Sin/3 10. 11. 12. (i) 1-i (1+ W) 15. (1 (1 i)* 14. 13. ( (-)(1+v) 17. 16 () 19. (1-) 21 20.
Transcribed Image Text:Remembering that any complex number can be written in the form re" by (9.4), we get Section 9 Euler's Formula 63 2 e2 = ri r2 e4(02+@2), e 21 22 (9.6) In words, to multiply two complex numbers, we multiply their absolute values and add their angles. To divide two complex numbers, we divide the absolute values and subtract the angles Example. Evaluate (1 + i)?/(1 - i). From Figure 5.1 we have 1 2et/4. We plot 1 - i in Figure 9.5 and find r2, 0=/4 (or +7/4), so 1-i= /2e-i#/4. Then (VEetr/a)2 V2e-i7/4=J%-in/s = /2e3i#/4 2 eir/2 (1+i)2 1-1 Figure 9.5 From Figure 9.6, we find =-1, y = 1, so (1i)2 1-1 =riy-1+i We could use degrees in this problem. By (9.6), we find that the angle of (1 i)2/(1-i) is 2(45°) - (-45°) 135° as in Figure 9.6 Figure 9.6 PROBLEMS, SECTION 9 Express the following complex numbers in the r + iy form. Try to visualize each complex number, using sketches as in the examples if necessary. The first twelve problems you should be able to do in your head (and maybe some of the others-try it!) Doing a problem quickly in your head saves time over using a computer. Remember that the point in doing problems like this is to gain skill in manipulating complex expressions, so a good study method is to do the problems by hand and use a computer to check your answers. 3. 93ri/2 e-2i -4mi - 2. ei/2 1. ei/4 (a/3)(344mi 4. 6. 5. 7. 3e2(1+i 2esri/6 9. 2e-i/2 /4 4e-Sin/3 10. 11. 12. (i) 1-i (1+ W) 15. (1 (1 i)* 14. 13. ( (-)(1+v) 17. 16 () 19. (1-) 21 20.
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A complex number in x+iy form is written as follows.

ecos(x)isin(x)
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