Uese De Moivre's formula.
(1-i)^10 = (√2 cis -π/4)^10 = 2^5 cis -10π/4 = 32 cis -π/2 = -32i
(1+i)^20 = (√2 cis π/4)^20 = 2^10 cis 5π = -1024
#1.) Write (1-i)^10 in a+bi form.
#2.) Write (1+i)^20 in a+bi form.
How do I do these? Thanks!
2 answers
1 - i = sqrt 2 [ cos 7pi/4 + i sin 7pi/4)
e^it = cos t + i sin t
so
1-i = 2^.5 e^(7 pi/4)
(1-i)^10 = 2^5 e^(70pi/4)
= 32 e^ (70 pi/4)
how many 2 pi circles in 70 pi/4
70/8 = 9.75
so 8 full circles and 3/4 of a circle of measure 2 pi
so
= 32 e^ (3 pi/2
cos 3 pi/2 = 0
sin 3 pi/2 = -1
so
32 ( 0 -1 i
- 32 i
e^it = cos t + i sin t
so
1-i = 2^.5 e^(7 pi/4)
(1-i)^10 = 2^5 e^(70pi/4)
= 32 e^ (70 pi/4)
how many 2 pi circles in 70 pi/4
70/8 = 9.75
so 8 full circles and 3/4 of a circle of measure 2 pi
so
= 32 e^ (3 pi/2
cos 3 pi/2 = 0
sin 3 pi/2 = -1
so
32 ( 0 -1 i
- 32 i