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Help me this one complex number question (1 Viewer)

upishcat

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I turned z1 and z2 into modulus argument form so I got z1 = cos(pi/2)+isin(pi/2) and z2= cos(pi/4)+isin(pi/4) but I dont know how to prove arg(z1+z2) as I had never seen this identity before.
 

Drdusk

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View attachment 27898
I turned z1 and z2 into modulus argument form so I got z1 = cos(pi/2)+isin(pi/2) and z2= cos(pi/4)+isin(pi/4) but I dont know how to prove arg(z1+z2) as I had never seen this identity before.
Why not just add the two complex numbers as they are? This gives you z1 + z2 = 'something', and then just find the argument using



Argument is calculated by taking tan inverse of the imaginary part over the real part of a complex number.
 

fan96

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Why not just add the two complex numbers as they are? This gives you z1 + z2 = 'something', and then just find the argument using



Argument is calculated by taking tan inverse of the imaginary part over the real part of a complex number.
The argument of a complex number satisfies



This is not the same as



because is not an inverse function of .

With it's clear that the above does not hold, as the function can't possibly have an output of .

In the case of this question, both numbers have modulus 1. If you draw a diagram you can probably see that the argument of the sum will be the average of the sum of the arguments (, and the origin form an isosceles triangle).
 
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Drdusk

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The argument of a complex number satisfies



This is not the same as



because is not an inverse function of .

With it's clear that the above does not hold, as the function can't possibly have an output of .

In the case of this question, both numbers have modulus 1. If you draw a diagram you can probably see that the argument of the sum will be the average of the sum of the arguments.
I waaas waiting for someone to say that imao.

It's such a small subtlety kinda negligible.
 

fan96

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It's such a small subtlety kinda negligible.
It's not negligible at all - it's the biggest defect in the "inverse" trig functions and it can completely throw off your calculations if you're not careful.

If instead we had then this method would've given a very wrong answer.

Suppose you're developing a navigation software with some sort of compass function.
Ignoring the fact that we don't actually live in two-dimensional space, if you just use the method discussed above, a bearing of 315 degrees is calculated as 135 degrees and now some poor bushwalker is probably very badly lost.
 

HeroWise

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You can do it geometrically too, Since its a rhobus yada yada
 

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