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12; tan u
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1> 23 and u 1 and u
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(c) P lies in the fourth quadrant; r Thus, z
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( 2)2
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2 22; tan u
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i sin 315 ).
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CHAPTER 15 Complex Numbers
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(d) P lies on the negative x axis and u Thus, z 3(cos 180 i sin 180 ). (e) P lies on the positive y axis and u Thus, z 4(cos 90 i sin 90 ).
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180 ; r
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2( 3)2 202
90 ; r
4. 1.3333 and u
(f) P lies in the third quadrant; r Thus, z 5(cos 233 8
2( 3)2
( 4)2
5; tan u
4/( 3)
233 8 .
i sin 233 8 ). u is not a special angle, so it must be approximated to get polar form.
15.13 Express each of the following complex numbers z in rectangular form: (a) 4(cos 240 i sin 240 ) (c) 3(cos 90 i sin 90 ) (b) 2(cos 315 i sin 315 ) (d) 5(cos 128 i sin 128 ).
(a) 4(cos 240 (b) 2(cos 315 (c) 3(cos 90 i sin 240 ) i sin 315 ) i sin 90 ) 4[ 1/2 2[1> 22 3[0 i(1)] i( 23>2)] i( 1> 22)] 3i 3.0785 3.9400i. Since 128 is not a special 2 22 2i 23 i 22
(d) 5(cos 128 i sin 128 ) < 5[ 0.6157 i(0.7880)] angle, its function values are approximated.
15.14 Prove: (a) The modulus of the product of two complex numbers is the product of their moduli, and the amplitude of the product is the sum of their amplitudes. (b) The modulus of the quotient of two complex numbers is the modulus of the dividend divided by the modulus of the divisor, and the amplitude of the quotient is the amplitude of the dividend minus the amplitude of the divisor.
Let z1 (a) z1z2 r1(cos r1(cos
1 1 1 1
i sin i sin cos
) and z2 sin
r2(cos
i sin
). cos sin )]
) ( r2(cos
i sin ) )] 2
r1r2[(cos r1r2[cos (
i(sin
i sin (
CHAPTER 15 Complex Numbers
r1( cos u1 r2( cos u2 i sin u1) i sin u2) r1( cos u1 r2( cos u2 i sin u1)( cos u2 i sin u2)( cos u2 i sin u2) i sin u2) i( sin u1 cos u2 sin 2u2 cos u1 sin u2)
r1 ( cos u1 cos u2 # r2 r1 r2 [ cos (u1 u2]
sin u1 sin u2) cos 2u2 i sin (u1 u2)]
15.15 Perform the indicated operations, giving the result in both polar and rectangular form. (a) (b) (c) (d) (e) (f) 5(cos 170 i sin 170 ) (cos 55 i sin 55 ) 2(cos 50 i sin 50 ) 3(cos 40 i sin 40 ) 6(cos 110 i sin 110 ) 1 (cos 212 i sin 212 ) 2 10(cos 305 i sin 305 ) 2(cos 65 i sin 65 ) 4(cos 220 i sin 220 ) 2(cos 40 i sin 40 ) 6(cos 230 i sin 230 ) 3(cos 75 i sin 75 )
(a) The modulus of the product is 5(1) 5 and the amplitude is 170 55 225 . In polar form the product is 5(cos 225 i sin 225 ), and in rectangular form the product is 5( 22>2 i 22>2) 522>2 5i 22>2. (b) The modulus of the product is 2(3) 6 and the amplitude is 50 40 90 . In polar form the product is 6(cos 90 i sin 90 ), and in rectangular form it is 6(0 (c) The modulus of the product is 6 A B
3 and the amplitude is 110
In polar form the product is 3(cos 322 0.6157i) 2.3640 1.8471i.
i sin 322 ), and in rectangular form it is approximately 3(0.7880
(d) The modulus of the quotient is 10/2 5 and the amplitude is 305 65 240 . In polar form the product is 5(cos 240 i sin 240 ), and in rectangular form it is 5( 1/2 5/2 5i 23>2. (e) The modulus of the quotient is 4/2 2 and the amplitude is 220 40 180 . In polar form the quotient is 2(cos 180 i sin 180 ), and in rectangular form it is 2( 1
i 23>2)
(f) The modulus of the quotient is 6/3 2 and the amplitude is 230 155 . 75 In polar form the quotient is 2(cos 155 i sin 155 ), and in rectangular form it is approximately 2( 0.9063 0.4226i) 1.8126 0.8452i.
15.16 Express each of the numbers in polar form, perform the indicated operation, and give the result in rectangular form. (a) ( 1 i 23)( 23 i) (b) (3 3i 23)( 2 2i 23) (c) (4 4i23) ( 223 2i)
(a) ( 1 (b) (3 i 23)( 23 3i 23) ( 2 i) 2(cos 120 4(cos 150 2i 23)
(d) 2 ( 23 i) (e) 6i ( 3 3i) (f) (1 i23)(1 i23)
i sin 120 ) 2(cos 30 i sin 150 ) 4 A 23>2 i sin 30 )
1 2i
2 23
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