## IrishBoy123 one year ago find

1. IrishBoy123

$\left| \dfrac{2e^{i \theta} - i}{ie^{i \theta}+2} \right|$

2. ganeshie8

we may use the property $$\left|\dfrac{z_1}{z_2}\right|=\dfrac{|z_1|}{|z_2|}$$ : $\left| \dfrac{2e^{i \theta} - i}{ie^{i \theta}+2} \right| = \dfrac{|2\cos\theta + i(2\sin\theta-1)|}{|2-\sin\theta+i\cos\theta|}=1$

3. IrishBoy123

yep!

4. UnkleRhaukus

|dw:1440667852028:dw|

5. IrishBoy123

|dw:1440668386653:dw|

6. IrishBoy123

@UnkleRhaukus

7. UnkleRhaukus

oh right, i got that second circle to the wrong scale.

8. UnkleRhaukus

|dw:1440668618259:dw|

9. 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?

10. UnkleRhaukus

the i is just a π/2 rotation

11. IrishBoy123

of course! see it

12. UnkleRhaukus

|dw:1440668948392:dw|

13. UnkleRhaukus

|dw:1440669146594:dw|

14. 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!

15. IrishBoy123

silly question but do i need to factor in the $$\pi/2$$ phase shift before comparing the numbers?

16. UnkleRhaukus

|dw:1440669675928:dw|

17. 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 $$\theta = 0$$

18. IrishBoy123

|dw:1440670317848:dw| and $$\theta = -\frac{\pi}{2}$$!!

19. IrishBoy123

|dw:1440670437299:dw| $$\theta = \pi$$

20. IrishBoy123

i can see! it's a bit blurry but i can see! thank you both

21. UnkleRhaukus

|dw:1440671876306:dw|