`7-7 De Moivre's Theorem561Section 7-7 De Moivre's TheoremDe Moivre's Theorem, n a Natural Number nth Roots of zWe now come to one of the great theorems in mathematics, De Moivre's theorem. Abraham De Moivre (1667­1754), of French birth, spent most of his life in London doing private tutoring, writing, and publishing mathematics. He belonged to many prestigious professional societies in England, Germany, and France and was a close friend of Isaac Newton. Using the polar form for a complex number, De Moivre established a theorem that still bears his name for raising complex numbers to natural number powers. More importantly, the theorem is the basis for the nth root theorem, which enables us to find all n nth roots of any complex number, real or imaginary.De Moivre's Theorem, n a Natural NumberWe start with Explore/Discuss 1 and generalize from this exploration.1By repeated use of the product formula for the exponential polar form rei , discussed in the last section, establish the following: (re i)2 r2e2 i 1. (x iy)2 3 i 3 2. (x iy) (re ) r3e3 i 3. (x iy)4 (re i)4 r4e4 i Based on forms 1­3, and for n a natural number, what do you think the polar form of (x iy)n would be?If you guessed that the polar form of (x iy)n is rnen i, you have arrived at De Moivre's Theorem, which we now state without proof. A full proof of the theorem for all natural numbers n requires a method of proof, called mathematical induction, which is discussed in Section 9-2.DE MOIVRE'S THEOREM If z x iy rei , and n is a natural number, then1EXAMPLEzn(xiy)n(rei )nrneniThe Natural Number Power of a Complex NumberUse De Moivre's theorem to find (1 gular form. i)10. Write the answer in exact rectan-15627 ADDITIONAL TOPICS IN TRIGONOMETRYSolution(1i)10( 2e45°i)10 ( 2)10e(10 32e450°i 45°)iConvert 1i to polar form.Use De Moivre's theorem. Change to rectangular form.32(cos 450° 32(0 32i i)i sin 450°)Rectangular formMATCHED PROBLEM1EXAMPLEUse De Moivre's theorem to find (1 and rectangular forms.i 3)5. Write the answer in exact polarThe Natural Number Power of a Complex NumberUse De Moivre's theorem to find ( tangular form. ( 3 i)6 (2e150°i)6 26e(6150°)i2Solution3i)6. Write the answer in exact recConvert form. 3 i to polarUse De Moivre's theorem. Change to rectangular form.64e900°i 64 (cos 900° 64 ( 1 64 [Note: 3 i must be a sixth root of i0) i sin 900°)Rectangular form64, since (3i)664.]MATCHED PROBLEM2Use De Moivre's theorem to find (1 and rectangular forms.i 3)4. Write the answer in exact polarnth Roots of zWe now consider roots of complex numbers. We say w is an nth root of z, n a z. For example, if w2 z, then w is a square root of z. natural number, if wn 3 z, then w is a cube root of z. And so on. If wIf z rei , then use De Moivre's theorem to show that r1/2e( square root of z and r1/3e( /3)i is a cube root of z./2)iis a2We can proceed in the same way as in Explore/Discuss 2 to show that r1/ne( is an nth root of rei , n a natural number:/n)i7-7 De Moivre's Theorem563[r1/ne(/n)i n](r1/n)nen( rei/n)iBut we can do even better than this. The nth-root theorem (Theorem 2) shows us how to find all the nth roots of a complex number.nTH-ROOT THEOREM For n a positive integer greater than 1,2r1/ne(/n k360°/n)ik0, 1, . . . , n1are the n distinct nth roots of rei , and there are no others.The proof of Theorem 2 is left to Problems 31 and 32 in Exercise 7-7.EXAMPLEFinding All Sixth Roots of a Complex NumberFind six distinct sixth roots of First write 1 1 i 3 1 i 3, and plot them in a complex plane.3Solutioni 3 in polar form: 2e120°iUsing the nth-root theorem, all six roots are given by 21/6e(120°/6 Thus, w1FIGURE 1k 360°/6)i21/6e(20°k60°)ik0, 1, 2, 3, 4, 521/6e(20° 21/6e(20° 21/6e(20° 21/6e(20° 21/6e(20° 21/6e(20°0 60°)i 1 60°)i 2 60°)i 3 60°)i 4 60°)i 5 60°)i21/6e20°i 21/6e80°i 21/6e140°i 21/6e200°i 21/6e260°i 21/6e320°iw2 w3 w4 w5 w6All roots are easily graphed in the complex plane after the first root is located. The root points are equally spaced around a circle of radius 21/6 at an angular increment of 60° from one root to the next (Fig. 1).MATCHED PROBLEM3Find five distinct fifth roots of 1 them in a complex plane.i. Leave the answers in polar form and plot5647 ADDITIONAL TOPICS IN TRIGONOMETRYEXAMPLESolving a Cubic EquationSolve x3 1 0. Write final answers in rectangular form, and plot them in a complex plane. x3 1 x34Solution0 1We see that x is a cube root of 1, and there are a total of three roots. To find the three roots, we first write 1 in polar form: 1 1e180°i 1 are given byUsing the nth-root theorem, all three cube roots of 11/3e(180°/3 Thus,FIGURE 2k360°/3)i1e(60°k120°)ik0, 1, 2w1 w2 w31e60°i le180°i le300°icos 60° cos 180° cos 300°i sin 60°1 2i3 2 1i sin 180° i sin 300° 1 2i3 2[Note: This problem can also be solved using factoring and the quadratic formula--try it.] The three roots are graphed in Figure 2. Solve x3 1 0. Write final answers in rectangular form, and plot them in a complex plane.MATCHED PROBLEM41. 32e300°i 3. w1 16 i16 3Answers to Matched Problems2. 16e(240°)i8i83 21/10e297°i 4. 1, 1 2 i 3 2 , 1 2 i 3 221/10e9°i, w221/10e81°i, w321/10e153°i, w421/10e225°i, w57-7 De Moivre's Theorem565EXERCISE 7-7 AIn Problems 1­6, use De Moivre's theorem to evaluate each. Leave answers in polar form. 1. (2e30°i)3 4. ( 2e15°i)8 2. (5e15°i)3 5. (1 i 3)3 3. ( 2e10°i)6 6. ( 3 i)826. (A) Show that 2 is a root of x3 8 0. How many other roots does the equation have? (B) The root 2 is located on a circle of radius 2 in the complex plane as indicated in the figure. Locate the other two roots of x3 8 0 on the figure and explain geometrically how you found their location. (C) Verify that each complex number found in part B is a root of x3 8 0.BIn Problems 7­12, find the value of each expression and write the final answer in exact rectangular form. (Verify the results in Problems 7­12 by evaluating each directly on a calculator.) 7. ( 10. ( 3 3 i)4 i)5 8. ( 1 11. 1 2 i)4 3 239. (1 i 12. 1 2i)8 3 23iFor n and z as indicated in Problems 13­18, find all nth roots of z. Leave answers in polar form. 13. z 15. z 17. z 8e30°i, n 81e 160°iIn Problems 27­30, solve each equation for all roots. Write final answers in polar and exact rectangular form 27. x3 64 27 0 0 28. x3 30. x3 64 27 0 03 4 514. z 16. z 18. z8e45°i, n 16e 190°i3 4 3,n,n i, n29. x3i, nFor n and z as indicated in Problems 19­24, find all nth roots of z. Write answers in polar form and plot in a complex plane. 19. z 21. z 23. z 8, n 3 4 20. z 22. z 24. z 1, n 8, n i, n 4 3 5C31. Show that [r1/ne(/n k360°/n)i n16, n i, n 6]reifor any natural number n and any integer k. 32. Show that r1/ne(/n k360°/n)i25. (A) Show that 1 i is a root of x4 4 0. How many other roots does the equation have? (B) The root 1 i is located on a circle of radius 2 in the complex plane as indicated in the figure. Locate the other three roots of x4 4 0 on the figure and explain geometrically how you found their location. (C) Verify that each complex number found in part B is a root of x4 4 0.is the same number for k0 and kn.In Problems 33­36, write answers in polar form. 33. Find all complex zeros for P(x) 34. Find all complex zeros for P(x) 35. Solve x5 3x5 x632. 1.1 i0 in the set of complex numbers. 0 in the set of complex numbers.36. Solve xIn Problems 37 and 38, write answers using exact rectangular forms. 37. Write P(x) 38. Write P(x) x6 x664 as a product of linear factors. 1 as a product of linear factors.`

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