Complex Variables - principal argument

In summary, the question asks for the principle argument of z when z = (sqrt(3) - i)^6. The solution involves finding theta such that exp(i*theta*6) = (sqrt(3)-i)^6, but the exact steps to solve this are unclear. After further consideration, the solution is found to be -pi/2.
  • #1
YoshiMoshi
232
10

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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  • #2
Never mind. I understand how to do it now.
 

Related to Complex Variables - principal argument

What is a principal argument in complex variables?

In complex variables, the principal argument is the unique angle in the complex plane between the positive real axis and the line connecting the origin to a given complex number. It is typically denoted by Arg(z) and is defined to be within the range of -π to π.

How is the principal argument related to the complex modulus?

The principal argument and the complex modulus of a complex number are related through the polar form of a complex number, z = r(cosθ + i sinθ). The principal argument is equal to the angle θ, while the complex modulus is equal to the absolute value of r.

What is the significance of the principal argument in complex analysis?

The principal argument is important in complex analysis because it allows us to define a continuous argument function for complex numbers, which is not possible for real numbers. It also allows us to define important functions such as the complex logarithm and the complex power function.

Can the principal argument of a complex number be negative?

Yes, the principal argument can be negative if the complex number is in the third or fourth quadrant of the complex plane. In these cases, the principal argument will be equivalent to the angle measured from the negative real axis instead of the positive real axis.

How is the principal argument used in solving complex equations?

The principal argument is used in solving complex equations by allowing us to express complex numbers in polar form, which makes it easier to perform calculations and find solutions. It also helps in visualizing complex numbers and their relationships in the complex plane.

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