# Finding least value of mod z in arg z diagrams

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#1
Why is the least value the sin part of z=r(cosx+isinx)? I don’t really understand how to derive the least value of mod z in arg z. Thank you for any help.
Last edited by Eris13696; 1 month ago
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1 month ago
#2
Have you drawn a diagram? The least value of mod z is the smallest distance between the half-line and the origin
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#3
(Original post by tej3141)
Have you drawn a diagram? The least value of mod z is the smallest distance between the half-line and the origin
Yes, I’m still not sure how to find it, if it doesn’t trouble you, can you explain it further?
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1 month ago
#4
(Original post by Eris13696)
Yes, I’m still not sure how to find it, if it doesn’t trouble you, can you explain it further?
Yh the 1st diagram is correct. I'm not 100% sure but i belive the question may have and error, it would make more sense if it where (z+3) =pi/3. Then you would find the shortest distance from the origin to the line by constructing a right angle triangle and using trig to get the answer (3•3^1/2)/2
Last edited by username5737602; 1 month ago
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1 month ago
#5
(Original post by Eris13696)
Yes, I’m still not sure how to find it, if it doesn’t trouble you, can you explain it further?
There are two diagrams here, the first is correctly of a half line. The second which has the triange and 3 marked on it and the line is extended is incorrect as arg() = pi/3 is a half line. The least value of |z| is simply 3 which occurs when z=3. As tej3141 says, the question may have an error. Equally, the solution could be messed up.

If the diagram was right (it isn't), you have a pi/6,pi/3,pi/2 right angled trangle, where the hypotenuse is 3. The shortest distance (side) is opposite the pi/3 angle so using simple trig its length is 3*sin(pi/3). It isn't really to do with the Euler formula for z
Last edited by mqb2766; 1 month ago
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