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CHAPTER 15 Complex Numbers
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15.22 Perform graphically the following operations. (a) (2 (b) (4 3i) 2i) (1 (2 4i) 3i) (c) (2 (d) (4 3i) 2i) (1 (2 4i) 3i)
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15.23 Express each of the following complex numbers in polar form. (a) 3 (b) 1 (c) (d) 22 3i 223 3 22(cos 45 2(cos 60 2i i 22 4(cos 210 2(cos 315 i sin 45 ) i sin 60 ) i sin 210 ) i sin 315 ) (e) (f) (g) (h) 8 2i 12 4 8(cos 180 2(cos 270 i sin 180 ) i sin 270 ) i sin 157 23 ) i sin 216 52 ) 23i
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15.24 Perform the indicated operation and express the results in the form a (a) 3(cos 25 (b) 4(cos 50 (c) (d) 4( cos 190 2( cos 70 12( cos 200 3( cos 350 i sin 25 ) 8(cos 200 i sin 50 ) 2(cos 100 i sin 190 ) i sin 70 ) i sin 200 ) i sin 350 ) 1 i sin 200 ) i sin 100 ) i 23 2i 12 22 4 23
12 22i 4i
15.25 Use the polar form in finding each of the following products and quotients, and express each result in the form a bi. (a) (1 (b) ( 1 i) A 22 i 22 B 2 22 4i) 8 23 8i (c) (d) 1 1 4 i i 4 23i 23 i i 2 23 2i
i23)( 4 23
15.26 Use De Moivre s theorem to evaluate each of the following and express each result in the form a (a) [2(cos 6 (b) 22(cos 75 (c) (1 (d) (1 (e) A 1/2 i)8 i)6 16 8i i23/2 B
i sin 6 )]
16 23
16i 2 23i
(f) A 23>2
i sin 75 )
A1 A 2
i 23 B
2i B 4 i) A 23
1 8 iB 3
i 23/2
i 23 B 3
15.27 Find all the indicated roots, expressing the results in the form a (a) The square root of i (b) The square roots of 1 (c) The cube roots of 8 i 23 Ans. 22>2 Ans. 26>2 Ans. 1 i 23,
bi unless tables would be needed to do so. 22>2 26>2 i 23 3 23>2 3i>2, 3i i sin 170 ), i 22>2 i 22>2
i 22>2, i 22>2, 2, 1 3i>2,
(d) The cube roots of 27i (e) The cube roots of 423 4i
Ans. 3 23>2 Ans. 2(cos 50 2(cos 290
i sin 50 ), 2(cos 170 i sin 290 )
CHAPTER 15 Complex Numbers
(f) The fifth roots of 1 (g) The sixth roots of
i 23 i
Ans. 22(cos 9 Ans. 22 (cos 25
i sin 9 ), 22(cos 81
i sin 81 ), etc. i sin 85 ), etc.
i sin 25 ), 22(cos 85
15.28 Find the tenth roots of 1 and show that the product of any two of them is again one of the tenth roots of 1. 15.29 Show that the reciprocal of any one of the tenth roots of 1 is again a tenth root of 1. 15.30 Denote either of the complex cube roots of (Prob. 15.19d) by v2 . 1 2 2 15.31 Show that (cos i sin )
and the other by
. Show that v2 1
cos n
i sin n .
15.32 Use the fact that the segments OS and P2P1 in Fig. 15.2(c) are equal to devise a second procedure for constructing the difference OS z1 z2 of two complex numbers z1 and z2.
APPENDIX 1
Geometry
A1.1 Introduction
Appendix 1 is a summary of basic geometry definitions, relations, and theorems. The purpose of this material is to provide information useful in solving problems in trigonometry.
A1.2 Angles
An angle is a figure determined by two rays having a common endpoint. An acute angle is an angle with a measure between 0 and 90 . A right angle is an angle with a measure of 90 , while an obtuse angle has a measure between 90 and 180 . When the sum of the measures of two angles is 90 , the angles are complementary. When the sum of the measures of two angles is 180 , the angles are supplementary. Two angles are equal when they have the same measure.
Fig. A1.1
If two lines intersect, the opposite angles are vertical angles. In Fig. A1.1(a), AED and BEC are vertical angles and CEA and BED are also a pair of vertical angles. When two angles have a common vertex and a common side between them, the angles are adjacent angles. In Fig. A1.1(b), PSQ and QSR are a pair of adjacent angles. If the exterior sides of two adjacent angles form a straight line, the angles form a linear pair. In Fig. A1.1(c), WXZ and ZXY are a linear pair.
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