show that $\sin(x)+\sin(x-120^{\circ})+\sin(x-240^{\circ})=0$

Question

ganeshie8 · Answer

my roomate and i are arguing about electric motors and im looking for some geometric explanation for this..

empty · Answer

what's true for x=0 is always true no matter how you rotate it

empty · Answer

$$e^{i x}(e^{i 2 \pi / 3} +e^{i 2 \pi 2/ 3} +e^{i 2 \pi 3/ 3}) =0$$

the real part of that

empty · Answer

imaginary part I mean, whatever, both are zero haha

michele_laino · Answer

it is a three phase system

rhr12 · Answer

x + 120 = t 

sin(t - 120) + sin(t) + sin(t + 120) = 0 
sin(t)cos(120) - sin(120)cos(t) + sin(t) + sin(t)cos(120) + sin(120)cos(t) = 0 
2 * cos(120) * sin(t) + sin(t) = 0 
sin(t) * (2 * cos(120) + 1) = 0 
sin(t) * (2 * (-1/2) + 1) = 0 
sin(t) * (-1 + 1) = 0 
sin(t) * 0 = 0 

So, sin(t) can equal pretty much anything 

t = x + 120 

sin(x + 120) can be anything 
x + 120 can be anything 
x can be anything 

There are no wrong solutions for this one.

rhr12 · Answer

That's all i got.

ganeshie8 · Answer

Yes he is saying delta/star and some bs which i have no idea about... but it makes sense to think in terms of roots of unity as Empty wa suggesting.. (if im interepreting correctly..)

michele_laino · Answer

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