Complex Variables - principal argument

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SUMMARY

The principal argument Arg z for the complex number z = (sqrt(3) - i)^6 can be determined using polar coordinates. The discussion highlights the need to express the complex number in the form of exp(i*theta) and find the appropriate angle theta. The user initially struggled with the relationship between sine and cosine values but later clarified their understanding of the problem. The correct approach involves recognizing that theta must satisfy both the cosine and sine equations derived from the complex number's rectangular form.

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  • Understanding of complex numbers and their polar representation
  • Familiarity with Euler's formula exp(i*theta)
  • Knowledge of trigonometric identities and their applications in complex analysis
  • Ability to manipulate and simplify complex expressions
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  • Study the derivation of polar coordinates for complex numbers
  • Learn about Euler's formula and its applications in complex analysis
  • Explore the properties of complex exponentiation and logarithms
  • Practice solving complex number problems involving principal arguments
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Students and educators in mathematics, particularly those focusing on complex analysis, as well as anyone seeking to deepen their understanding of complex numbers and their properties.

YoshiMoshi
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Homework Statement



Find the principle argument Arg z when

z = (sqrt(3) - i)^6

Homework Equations

The Attempt at a Solution



I'm sorry to say that I'm not sure how to solve this problem. It's my understanding that what this question is basically asking me to do is find theta such that

exp(i*theta*6) = (sqrt(3)-i)^6

However I'm not sure how to solve this.

cos(theta) = sqrt(3)
sin(theta) = -i

In order for sin(theta) = -1, then theta must be equal to -pi/2. However cos(-pi/2) =/= sqrt(3). I seem to not understand what this question is asking me exactly or how to solve this problem.

Thanks for any help.
 
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Never mind. I understand how to do it now.
 

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