Help me this one complex number question (1 Viewer)

[upishcat]

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]

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]

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).

 

[Drdusk]

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]

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.

 

[upishcat]

Thank you very much for the help guys.

 

[HeroWise]

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