Solving for the Argument of Complex Numbers: arg(z^4 - 1i√3) 4π/3
Understanding how to solve for the argument of complex numbers in specific scenarios is an essential skill in advanced mathematics. This article will guide you through the process of solving the equation arg(z^4 - 1i√3) 4π/3. We will break down the steps and explain the reasoning behind each move to ensure clarity and comprehension.
Step 1: Calculate arg(-1i√3)
The first step in solving the equation is to find the argument of the complex number -1i√3. We can break this down as follows:
First, recognize that -1i√3 is a complex number of the form a bi, where a 0 and b -√3. Knowing this, we can calculate the argument (angle) using the formula: Note that the argument of a complex number r(cosθ isinθ) can be found using θ atan(b/a) and the correct quadrant. For the complex number -1i√3, the angle is in the fourth quadrant, where the tangent is negative. Therefore, the argument is 2π/3.Thus, we have:
arg(-1i√3) 2π/3
Step 2: Analyze the Given Equation
Using the information provided, we are told that:
arg(z^4 - 1i√3) 4π/3
From this, we can infer:
arg(z^4) * arg(-1i√3) 4π/3
Substituting the value of arg(-1i√3) from the previous step:
arg(z^4) * (2π/3) 4π/3
Solving for arg(z^4):
arg(z^4) 2π/3
Step 3: Determine the Argument of z
Knowing that the argument of a complex number raised to a power is multiplied by that power, we can solve for the argument of z:
arg(z^4) 4 * arg(z)
Substituting the value of arg(z^4) from the previous step:
2π/3 4 * arg(z)
Solving for arg(z):
arg(z) (2π/3) / 4 π/6
Understanding the Result
The argument of z is:
arg(z) π/6
This means that the complex number z lies in the first quadrant at an angle of π/6 from the positive real axis.
Why We Can't Determine the Magnitude of z?
It's important to acknowledge that in this problem, we cannot determine the magnitude (or modulus) of z from the given information. The modulus of a complex number is given by its distance from the origin in the complex plane, and it requires additional information to be determined.
Conclusion
By carefully following the steps and applying the principles of complex numbers and their arguments, we have solved the equation arg(z^4 - 1i√3) 4π/3. We found that the argument of z is π/6. However, the modulus of z remains undetermined without additional information.