Solve Complex Numbers Homework: Arg (z) = pi/3

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Homework Help Overview

The problem involves finding the value of x in the complex number z = x + (x+1)i such that the argument of z, Arg(z), equals π/3. The discussion centers around understanding the relationship between the components of the complex number and the angle in the complex plane.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants explore the relationship between the argument of a complex number and its components, questioning the setup and definitions used in the equations. There are attempts to express the argument in terms of x and to clarify the correct application of the tangent function.

Discussion Status

The discussion is active, with participants providing guidance on how to approach the problem geometrically and algebraically. There is an ongoing exploration of the correct interpretation of the tangent function in relation to the argument of z.

Contextual Notes

Participants assume x is a real number and discuss the implications of this assumption in the context of the complex plane. There is a focus on ensuring that the definitions and relationships used are accurate, particularly regarding the tangent function.

missmerisha
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Homework Statement


If z = x + ( x+1) i, find the value of x for which Arg (z) = pi/3


Homework Equations





The Attempt at a Solution


( x+1/x) = pi/3
x = 3/( pi -3)

Answer: ( 3) ^(1/2) + 1 divided by 2
 
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missmerisha said:
( x+1/x) = pi/3
x = 3/( pi -3)
Where did you get these equations from? Remember that [itex]\arg z = \theta[/itex] is equivalent to [itex]z = |z|(\cos \theta + i \sin \theta)[/itex].
 
I'm working out the arg of z in term of x
tan ( y/ x) = ( x+1)/ x = pi/3
 
I assume x is a real number, correct?

It might help you to draw the point z in the complex plane. What is the horizontal component of z? What is the vertical component? Where is the angle arg(z)? You should have a right triangle with one of the angles equal to pi/3.

Once you've drawn this, you can use what you know about this triangle to write the equation for x. (pi/3 rad = 30 deg, in case you forgot that.)
 
Oops - your second post came in while I was responding.

Be careful - arg(z) = pi/3 is the angle, so tan(y/x) = pi/3 is wrong.

(You've just got it backwards - recall how tan() is defined and you're there.)
 
so it's
tan ( pi/3 ) = ( x+ 1 / x)
 
missmerisha said:
isn't the equation just
tan ( (x+1)/x) = pi/3 ?

Nope. What kind of number goes into the argument of a trig function? (You're really close, you just need to remember the definition of the tan function a little better.)
 
missmerisha said:
so it's
tan ( pi/3 ) = ( x+ 1 / x)

We crossed messages again ...

Yes! tan(angle) = y/x, so now you can solve the equation you just wrote.
 
don't worry, I've got it. Thank you!
 
  • #10
See my post.
 

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