IrishBoy123
  • IrishBoy123
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Mathematics
  • Stacey Warren - Expert brainly.com
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katieb
  • katieb
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IrishBoy123
  • IrishBoy123
\[\left| \dfrac{2e^{i \theta} - i}{ie^{i \theta}+2} \right|\]
ganeshie8
  • ganeshie8
we may use the property \(\left|\dfrac{z_1}{z_2}\right|=\dfrac{|z_1|}{|z_2|}\) : \[\left| \dfrac{2e^{i \theta} - i}{ie^{i \theta}+2} \right| = \dfrac{|2\cos\theta + i(2\sin\theta-1)|}{|2-\sin\theta+i\cos\theta|}=1\]
IrishBoy123
  • IrishBoy123
yep!

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UnkleRhaukus
  • UnkleRhaukus
|dw:1440667852028:dw|
IrishBoy123
  • IrishBoy123
|dw:1440668386653:dw|
IrishBoy123
  • IrishBoy123
@UnkleRhaukus
UnkleRhaukus
  • UnkleRhaukus
oh right, i got that second circle to the wrong scale.
UnkleRhaukus
  • UnkleRhaukus
|dw:1440668618259:dw|
IrishBoy123
  • IrishBoy123
my little brain is whirring here so for the denominator you take the unit circle and shift it right because of the \(+2\) and the fact it is also \(\times i \) only changes the sin and the cos round and does not change the actual number itself?
UnkleRhaukus
  • UnkleRhaukus
the i is just a π/2 rotation
IrishBoy123
  • IrishBoy123
of course! see it
UnkleRhaukus
  • UnkleRhaukus
|dw:1440668948392:dw|
UnkleRhaukus
  • UnkleRhaukus
|dw:1440669146594:dw|
IrishBoy123
  • IrishBoy123
i'm miles behind you on the triangles but i can see where you came out, therefore i "should" be able to figure it out in a while. my first instinct is always to draw things so i will study these carefully. i am thinking of sitting through these https://www.youtube.com/watch?v=bAJsyojpekI&list=PLkfmuPOLLntiM_HjG-FSUTyINMfgVszca&index=1 i love this stuff but it's been so long and i'm trying to find my way around. drawing/seeing it for me is really important. so thanks!
IrishBoy123
  • IrishBoy123
silly question but do i need to factor in the \(\pi/2\) phase shift before comparing the numbers?
UnkleRhaukus
  • UnkleRhaukus
|dw:1440669675928:dw|
IrishBoy123
  • IrishBoy123
\[\left| \frac{2e^{iθ}−i}{e^{i(θ+\frac{\pi}{2})}+2} \right|\] |dw:1440670064162:dw| yeah, i think i'm getting there my crap drawing is for the specialised case \(\theta = 0\)
IrishBoy123
  • IrishBoy123
|dw:1440670317848:dw| and \(\theta = -\frac{\pi}{2} \)!!
IrishBoy123
  • IrishBoy123
|dw:1440670437299:dw| \(\theta = \pi\)
IrishBoy123
  • IrishBoy123
i can see! it's a bit blurry but i can see! thank you both
UnkleRhaukus
  • UnkleRhaukus
|dw:1440671876306:dw|