2 wiskunde opdrachten

[bootje]

Z^4 = -8 + 8 i wortel3
Z = 4dewortel(–8 + 8 i wortel3)
= (–8 + 8Iwortel3)^1/4

r (modulus) = wortel(a² + b²)
= wortel(64 + (64.3))
=wortel256 = 16

tg (tangens) van argument = b/a
= 8wortel3/-8 = -wortel3
argument = 2 ð/3

Z^4 = 16 (cos (2ð/3 + 2kð) + i sin (2ð/3 + 2kð))
Z = 4dewortel16 (cos (ð/6 + kð/2) + i sin (ð/6 + kð/2))

k = 0--> 2 (cos ð/6 + i sin ð/6) = 2(wortel3/2 + i ½) = (wortel3) + i

k = 1 --> 2 (cos 4ð/6 + i sin 4ð/6) = 2(-1/2 + i (wortel3)/2
= -1 + i wortel3

k = 2 --> 2 (cos 7ð/6 + i sin 7ð/6) = 2 (-(wortel3)/2 + i ½)
= -(wortel3)- i

k = 3 --> 2 (cos 10ð/6 + i sin 10ð/6) = 2 ( ½ - i wortel3)
= 1 – i wortel3



cos = cosinus
sin = sinus
ð = pi