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Transcript
7-7 De Moivre’s Theorem 561 Section 7-7 De Moivre’s Theorem De Moivre’s Theorem, n a Natural Number nth Roots of z We now come to one of the great theorems in mathematics, De Moivre’s theorem. Abraham De Moivre (1667–1754), 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 Number We start with Explore/Discuss 1 and generalize from this exploration. 1 By repeated use of the product formula for the exponential polar form rei, discussed in the last section, establish the following: 1. (x ⫹ iy)2 ⫽ (rei)2 ⫽ r2e2i 2. (x ⫹ iy)3 ⫽ (rei)3 ⫽ r3e3i 3. (x ⫹ iy)4 ⫽ (rei)4 ⫽ r4e4i Based on forms 1–3, 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 rneni, 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, then 1 EXAMPLE 1 zn ⫽ (x ⫹ iy)n ⫽ (rei)n ⫽ rneni The Natural Number Power of a Complex Number Use De Moivre’s theorem to find (1 ⫹ i)10. Write the answer in exact rectangular form. 562 7 ADDITIONAL TOPICS IN TRIGONOMETRY Solution (1 ⫹ i)10 ⫽ (兹2e45°i)10 Convert 1 ⫹ i to polar form. ⫽ (兹2)10e(10 ⴢ 45°)i Use De Moivre’s theorem. ⫽ 32e Change to rectangular form. 450°i ⫽ 32(cos 450° ⫹ i sin 450°) ⫽ 32(0 ⫹ i) ⫽ 32i MATCHED PROBLEM 1 EXAMPLE 2 Solution Rectangular form Use De Moivre’s theorem to find (1 ⫹ i兹3)5. Write the answer in exact polar and rectangular forms. The Natural Number Power of a Complex Number Use De Moivre’s theorem to find (⫺兹3 ⫹ i)6. Write the answer in exact rectangular form. (⫺ 兹3 ⫹ i)6 ⫽ (2e150°i)6 Convert ⫺兹3 ⫹ i to polar form. ⫽ 26e(6 ⴢ 150°)i Use De Moivre’s theorem. ⫽ 64e900°i Change to rectangular form. ⫽ 64 (cos 900° ⫹ i sin 900°) ⫽ 64 (⫺1 ⫹ i0) ⫽ ⫺64 Rectangular form [Note: ⫺兹3 ⫹ i must be a sixth root of ⫺64, since (⫺兹3 ⫹ i)6 ⫽ ⫺64.] MATCHED PROBLEM 2 Use De Moivre’s theorem to find (1 ⫺ i兹3)4. Write the answer in exact polar and rectangular forms. nth Roots of z We now consider roots of complex numbers. We say w is an nth root of z, n a natural number, if wn ⫽ z. For example, if w2 ⫽ z, then w is a square root of z. If w3 ⫽ z, then w is a cube root of z. And so on. If z ⫽ rei, then use De Moivre’s theorem to show that r1/2e(/2)i is a square root of z and r1/3e(/3)i is a cube root of z. 2 We can proceed in the same way as in Explore/Discuss 2 to show that r1/ne(/n)i is an nth root of rei, n a natural number: 7-7 De Moivre’s Theorem 563 [r1/ne(/n)i]n ⫽ (r1/n)nen(/n)i ⫽ rei But 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, 2 r1/ne(/n⫹k360°/n)i k ⫽ 0, 1, . . . , n ⫺ 1 are 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. EXAMPLE 3 Solution Finding All Sixth Roots of a Complex Number Find six distinct sixth roots of ⫺1 ⫹ i兹3, and plot them in a complex plane. First write ⫺1 ⫹ i兹3 in polar form: ⫺1 ⫹ i兹3 ⫽ 2e120°i Using the nth-root theorem, all six roots are given by 21/6e(120°/6⫹k 360°/6)i ⫽ 21/6e(20°⫹k60°)i k ⫽ 0, 1, 2, 3, 4, 5 Thus, w1 ⫽ 21/6e(20°⫹0ⴢ60°)i ⫽ 21/6e20°i FIGURE 1 w2 ⫽ 21/6e(20°⫹1ⴢ60°)i ⫽ 21/6e80°i w3 ⫽ 21/6e(20°⫹2ⴢ60°)i ⫽ 21/6e140°i w4 ⫽ 21/6e(20°⫹3ⴢ60°)i ⫽ 21/6e200°i w5 ⫽ 21/6e(20°⫹4ⴢ60°)i ⫽ 21/6e260°i w6 ⫽ 21/6e(20°⫹5ⴢ60°)i ⫽ 21/6e320°i All 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 PROBLEM 3 Find five distinct fifth roots of 1 ⫹ i. Leave the answers in polar form and plot them in a complex plane. 564 7 ADDITIONAL TOPICS IN TRIGONOMETRY EXAMPLE 4 Solution Solving a Cubic Equation Solve x3 ⫹ 1 ⫽ 0. Write final answers in rectangular form, and plot them in a complex plane. x3 ⫹ 1 ⫽ 0 x3 ⫽ ⫺1 We 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 Using the nth-root theorem, all three cube roots of ⫺1 are given by 11/3e(180°/3⫹k360°/3)i ⫽ 1e(60°⫹k120°)i k ⫽ 0, 1, 2 Thus, FIGURE 2 w1 ⫽ 1e60°i ⫽ cos 60° ⫹ i sin 60° ⫽ 1 兹3 ⫹i 2 2 w2 ⫽ le180°i ⫽ cos 180° ⫹ i sin 180° ⫽ ⫺1 w3 ⫽ le300°i ⫽ cos 300° ⫹ i sin 300° ⫽ 1 兹3 ⫺i 2 2 [Note: This problem can also be solved using factoring and the quadratic formula—try it.] The three roots are graphed in Figure 2. MATCHED PROBLEM 4 Solve x3 ⫺ 1 ⫽ 0. Write final answers in rectangular form, and plot them in a complex plane. Answers to Matched Problems 1. 32e300°i ⫽ 16 ⫺ i16兹3 2. 16e(⫺240°)i ⫽ ⫺8 ⫹ i8兹3 3. w1 ⫽ 21/10e9°i, w2 ⫽ 21/10e81°i, w3 ⫽ 21/10e153°i, w4 ⫽ 21/10e225°i, w5 ⫽ 21/10e297°i 1 兹3 1 兹3 4. 1, ⫺ ⫹ i ,⫺ ⫺i 2 2 2 2 7-7 De Moivre’s Theorem 565 26. (A) Show that ⫺2 is a root of x3 ⫹ 8 ⫽ 0. How many other roots does the equation have? EXERCISE 7-7 (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. A In Problems 1–6, use De Moivre’s theorem to evaluate each. Leave answers in polar form. 1. (2e30°i)3 2. (5e15°i)3 3. (兹2e10°i)6 4. (兹2e15°i)8 5. (1 ⫹ i兹3)3 6. (兹3 ⫹ i)8 (C) Verify that each complex number found in part B is a root of x3 ⫹ 8 ⫽ 0. B In Problems 7–12, find the value of each expression and write the final answer in exact rectangular form. (Verify the results in Problems 7–12 by evaluating each directly on a calculator.) 7. (⫺兹3 ⫺ i)4 10. (⫺兹3 ⫹ i)5 9. (1 ⫺ i)8 8. (⫺1 ⫹ i)4 11. 冢 1 兹3 ⫺ ⫹ i 2 2 冣 3 12. 冢 1 兹3 ⫺ ⫺ i 2 2 冣 3 For n and z as indicated in Problems 13–18, find all nth roots of z. Leave answers in polar form. 13. z ⫽ 8e30°i, n ⫽ 3 15. z ⫽ 81e ,n⫽4 60°i 17. z ⫽ 1 ⫺ i, n ⫽ 5 14. z ⫽ 8e45°i, n ⫽ 3 16. z ⫽ 16e ,n⫽4 90°i 18. z ⫽ ⫺1 ⫹ i, n ⫽ 3 In Problems 27–30, solve each equation for all roots. Write final answers in polar and exact rectangular form 27. x3 ⫹ 64 ⫽ 0 28. x3 ⫺ 64 ⫽ 0 29. x3 ⫺ 27 ⫽ 0 30. x3 ⫹ 27 ⫽ 0 For n and z as indicated in Problems 19–24, find all nth roots of z. Write answers in polar form and plot in a complex plane. C 19. z ⫽ 8, n ⫽ 3 20. z ⫽ 1, n ⫽ 4 31. Show that 21. z ⫽ ⫺16, n ⫽ 4 22. z ⫽ ⫺8, n ⫽ 3 23. z ⫽ i, n ⫽ 6 24. z ⫽ ⫺i, n ⫽ 5 25. (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 兹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. [r1/ne(/n⫹k360°/n)i]n ⫽ rei for any natural number n and any integer k. 32. Show that r1/ne(/n⫹k360°/n)i is the same number for k ⫽ 0 and k ⫽ n. In Problems 33–36, write answers in polar form. 33. Find all complex zeros for P(x) ⫽ x5 ⫺ 32. 34. Find all complex zeros for P(x) ⫽ x6 ⫹ 1. 35. Solve x5 ⫹ 1 ⫽ 0 in the set of complex numbers. 36. Solve x3 ⫺ i ⫽ 0 in the set of complex numbers. In Problems 37 and 38, write answers using exact rectangular forms. 37. Write P(x) ⫽ x6 ⫹ 64 as a product of linear factors. 38. Write P(x) ⫽ x6 ⫺ 1 as a product of linear factors.