Números Complexos: Exercícios Resolvidos e Problemas Complementares, Exercícios de Matemática

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1.52. Prove that for m ¼ 2 , 3 ,...

sin

p

m

sin

2 p

m

sin

3 p

m

" " " sin

(m # 1) p

m

m

2 m#^1

Solution

The roots of zm^ ¼ 1 are z ¼ 1, e^2 pi=m, e^4 pi=m,... , e2 (m#1)^ pi=m. Then, we can write zm^ # 1 ¼ (z # 1)(z # e^2 pi=m)(z # e^4 pi=m) " " " (z # e2 (m#1)^ pi=m) Dividing both sides by z # 1 and then letting z ¼ 1 [realizing that (zm^ # 1)=(z # 1) ¼ 1 þ z þ z^2 þ " " " þ zm#^1 ], we find m ¼ (1 # e^2 pi=m)(1 # e^4 pi=m) " " " (1 # e2 (m#1)^ pi=m) (1) Taking the complex conjugate of both sides of (1) yields m ¼ (1 # e#^2 pi=m)(1 # e#^4 pi=m) " " " (1 # e#2 (m#1)^ pi=m) (2 ) Multiplying (1) by (2 ) using (1 # e^2 k^ pi=m)(1 # e#^2 k^ pi=m) ¼ 2 # 2 cos(2 k p=m), we have m^2 ¼ 2 m#^1 1 # cos 2 p m

1 # cos 4 p m

" " " 1 # cos 2 (m # 1) p m

Since 1 # cos(2 k p=m) ¼ 2 sin^2 (k p=m), (3 ) becomes m^2 ¼ 2 2 m#^2 sin^2 p m sin^2 2 p m " " " sin^2 (m # 1) p m

Then, taking the positive square root of both sides yields the required result.

SUPPLEMENTARY PROBLEMS

Fundamental Operations with Complex Numbers 1.53. Perform each of the indicated operations: (a) (4 # 3 i) þ (2 i # 8), (d) (i # 2 )f2 (1 þ i) # 3 (i # 1)g, (g) (2 þ i)(3 # 2 i)(1 þ 2 i) (1 # i)^2 (b) 3 (# 1 þ 4 i) # 2 (7 # i), (e) 2 # 3 i 4 # i , (h) (2 i # 1)^2

1 # i þ 2 # i 1 þ i

(c) (3 þ 2 i)(2 # i), (f ) (4 þ i)(3 þ 2 i)(1 # i) (i) i^4 þ i^9 þ i^16 2 # i^5 þ i^10 # i^15 1.54. Suppose z 1 ¼ 1 # i, z 2 ¼ # 2 þ 4 i, z 3 ¼ ffiffiffi 3 p

2 i. Evaluate each of the following:

(a) z^21 þ 2 z 1 # 3 (d) jz 1 z 2 þ z 2 z 1 j (g) (z 2 þ z 3 )(z 1 # z 3 ) (b) j 2 z 2 # 3 z 1 j^2 (e) z 1 þ z 2 þ 1 z 1 # z 2 þ i (h) jz^21 þ z^22 j^2 þ jz^23 # z^22 j^2 (c) (z 3 # z 3 )^5 (f )

z 3 z 3 þ z 3 z 3

(i) Ref 2 z^31 þ 3 z^22 # 5 z^23 g

32 CHAPTER 1 Complex Numbers

1.55. Prove that (a) (z 1 z 2 ) ¼ z 1 z 2 , (b) (z 1 z 2 z 3 ) ¼ z 1 z 2 z 3. Generalize these results. 1.56. Prove that (a) (z 1 =z 2 ) ¼ z 1 =z 2 , (b) jz 1 =z 2 j ¼ jz 1 j=jz 2 j if z 2 = 0. 1.57. Find real numbers x and y such that 2 x # 3 iy þ 4 ix # 2 y # 5 # 10 i ¼ (x þ y þ 2 ) # (y # x þ 3 )i. 1.58. Prove that (a) Refzg ¼ (z þ z)=2 , (b) Imfzg ¼ (z # z)= 2 i. 1.59. Suppose the product of two complex numbers is zero. Prove that at least one of the numbers must be zero. 1.60. Let w ¼ 3 iz # z^2 and z ¼ x þ iy. Find jwj^2 in terms of x and y. Graphical Representation of Complex Numbers. Vectors. 1.61. Perform the indicated operations both analytically and graphically. (a) (2 þ 3 i) þ (4 # 5 i) (d) 3 (1 þ i) þ 2 (4 # 3 i) # (2 þ 5 i) (b) (7 þ i) # (4 # 2 i) (e) 12 (4 # 3 i) þ 32 (5 þ 2 i) (c) 3 (1 þ 2 i) # 2 (2 # 3 i) 1.62. Let z 1 , z 2 , and z 3 be the vectors indicated in Fig. 1-40. Construct graphically: (a) 2 z 1 þ z 3 (b) (z 1 þ z 2 ) þ z 3 (c) z 1 þ (z 2 þ z 3 ) (d) 3 z 1 # 2 z 2 þ 5 z 3 (e) 13 z 2 # 34 z 1 þ 23 z 3 1.63. Let z 1 ¼ 4 # 3 i and z 2 ¼ # 1 þ 2 i. Obtain graphically and analytically (a) jz 1 þ z 2 j, (b) jz 1 # z 2 j, (c) z 1 # z 2 , (d) j 2 z 1 # 3 z 2 # 2 j. 1.64. The position vectors of points A, B, and C of triangle ABC are given by z 1 ¼ 1 þ 2 i, z 2 ¼ 4 # 2 i, and z 3 ¼ 1 # 6 i, respectively. Prove that ABC is an isosceles triangle and find the lengths of the sides. 1.65. Let z 1 , z 2 , z 3 , z 4 be the position vectors of the vertices for quadrilateral ABCD. Prove that ABCD is a parallelogram if and only if z 1 # z 2 # z 3 þ z 4 ¼ 0. 1.66. Suppose the diagonals of a quadrilateral bisect each other. Prove that the quadrilateral is a parallelogram. 1.67. Prove that the medians of a triangle meet in a point. 1.68. Let ABCD be a quadrilateral and E, F, G, H the midpoints of the sides. Prove that EFGH is a parallelogram. 1.69. In parallelogram ABCD, point E bisects side AD. Prove that the point where BE meets AC trisects AC. 1.70. The position vectors of points A and B are 2 þ i and 3 # 2 i, respectively. (a) Find an equation for line AB. (b) Find an equation for the line perpendicular to AB at its midpoint. 1.71. Describe and graph the locus represented by each of the following: (a) jz # ij ¼ 2 , (b) jz þ 2 ij þ jz # 2 ij ¼ 6, (c) jz # 3 j # jz þ 3 j ¼ 4, (d) z(z þ 2 ) ¼ 3 , (e) Imfz^2 g ¼ 4. 1.72. Find an equation for (a) a circle of radius 2 with center at (#3 , 4), (b) an ellipse with foci at (0, 2 ) and (0, #2 ) whose major axis has length 10. z (^2) z 1 z 3 x y Fig. 1-

CHAPTER 1 Complex Numbers 33

De Moivre’s Theorem 1.89. Evaluate each of the following: (a) (5 cis 2 0 8 )(3 cis 40 8 ) (b) (2 cis 50 8 )^6

(c)

(8 cis 40 8 )^3

(2 cis 60 8 )^4

(d)

(3 e^ pi=^6 )(2 e#^5 pi=^4 )(6e^5 pi=^3 )

(4e^2 pi=^3 )^2

(e)

ffiffiffi

p

# i

ffiffiffi

p

þ i

1 þ i

1 # i

1.90. Prove that (a) sin 3 u ¼ 3 sin u # 4 sin^3 u, (b) cos 3 u ¼ 4 cos^3 u # 3 cos u. 1.91. Prove that the solutions of z^4 # 3 z^2 þ 1 ¼ 0 are given by z ¼ 2 cos 3 6 8 , 2 cos 72 8 , 2 cos 2 16 8 , 2 cos 2 52 8. 1.92. Show that (a) cos 3 6 8 ¼ ( ffiffiffi 5 p þ 1)=4, (b) cos 72 8 ¼ ( ffiffiffi 5 p

1)=4. [Hint: Use Problem 1.91.]

1.93. Prove that (a) sin 4 u=sin u ¼ 8 cos^3 u # 4 cos u ¼ 2 cos 3 u þ 2 cos u (b) cos 4 u ¼ 8 sin^4 u # 8 sin^2 u þ 1 1.94. Prove De Moivre’s theorem for (a) negative integers, (b) rational numbers. Roots of Complex Numbers 1.95. Find each of the indicated roots and locate them graphically. (a) ( ffiffiffi 3 p

2 i)^1 =^2 , (b) (# 4 þ 4 i)^1 =^5 , (c) (2 þ 2

ffiffiffi 3 p i)^1 =^3 , (d) (# 16 i)^1 =^4 , (e) (64)^1 =^6 , (f) (i)^2 =^3. 1.96. Find all the indicated roots and locate them in the complex plane. (a) Cube roots of 8, (b) square roots of 4 ffiffiffi 2 p þ 4 ffiffiffi 2 p i, (c) fifth roots of # 16 þ 16 ffiffiffi 3 p i, (d) sixth roots of #2 7i. 1.97. Solve the equations (a) z^4 þ 81 ¼ 0, (b) z^6 þ 1 ¼ ffiffiffi 3 p i. 1.98. Find the square roots of (a) 5 # 12 i, (b) 8 þ 4 ffiffiffi 5 p i. 1.99. Find the cube roots of # 11 # 2 i. Polynomial Equations 1.100. Solve the following equations, obtaining all roots: (a) 5z^2 þ 2 z þ 10 ¼ 0, (b) z^2 þ (i # 2 )z þ (3 # i) ¼ 0. 1.101. Solve z^5 # 2 z^4 # z^3 þ 6 z # 4 ¼ 0. 1.102. (a) Find all the roots of z^4 þ z^2 þ 1 ¼ 0 and (b) locate them in the complex plane. 1.103. Prove that the sum of the roots of a 0 zn^ þ a 1 zn#^1 þ a 2 zn#^2 þ " " " þ an ¼ 0 where a 0 =0 taken r at a time is (#1)r^ ar =a 0 where 0 , r , n. 1.104. Find two numbers whose sum is 4 and whose product is 8. The nth Roots of Unity 1.105. Find all the (a) fourth roots, (b) seventh roots of unity, and exhibit them graphically. 1.106. (a) Prove that 1 þ cos 72 8 þ cos 144 8 þ cos 2 16 8 þ cos 2 88 8 ¼ 0. (b) Give a graphical interpretation of the result in (a). 1.107. Prove that cos 3 6 8 þ cos 72 8 þ cos 108 8 þ cos 144 8 ¼ 0 and interpret graphically.

CHAPTER 1 Complex Numbers 35