Why Does Arg(z) of a Complex Number Differ in Solutions?

In summary, the conversation is discussing the expression of arg(z) and polar form for a given complex number. The participant shares their attempt at a solution, but is unsure if their answer is correct due to conflicting answer keys. The expert suggests that the argument may be in the third quadrant, but also questions if the question was asking for the principal argument.
  • #1
charmedbeauty
271
0

Homework Statement



express the arg(z) and polar form of

([itex]1/\sqrt{2}[/itex]) - ([itex]i/\sqrt{2}[/itex])


Homework Equations





The Attempt at a Solution



Ok so I did [itex]\sqrt{(1/\sqrt{2})^{2}+(1/\sqrt{2})^{2}}[/itex] = 1

so tan[itex]^{-1}[/itex](1) = [itex]\pi/4[/itex] so arg(z)=5[itex]\pi[/itex][itex]/4[/itex]

but they had the answer as [itex]-3\pi/4[/itex]

Am I wrong or are they because shouldn't the arg(z) lie in the third quad.??
 
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  • #2
A better question for you is why do those numbers represent the same angle
 
  • #3
Both [itex]5\pi/4[/itex] and [itex]-3\pi/4[/itex] are in the third quadrant.
 
  • #4
charmedbeauty said:

Homework Statement



express the arg(z) and polar form of

([itex]1/\sqrt{2}[/itex]) - ([itex]i/\sqrt{2}[/itex])


Homework Equations





The Attempt at a Solution



Ok so I did [itex]\sqrt{(1/\sqrt{2})^{2}+(1/\sqrt{2})^{2}}[/itex] = 1

so tan[itex]^{-1}[/itex](1) = [itex]\pi/4[/itex] so arg(z)=5[itex]\pi[/itex][itex]/4[/itex]

but they had the answer as [itex]-3\pi/4[/itex]

Am I wrong or are they because shouldn't the arg(z) lie in the third quad.??

Were they asking for the principal argument? i.e. Arg(z)? Arg(z) is defined to be in the range of [itex](-\pi,\pi][/itex]
 

FAQ: Why Does Arg(z) of a Complex Number Differ in Solutions?

What is polar form and how is it different from rectangular form?

Polar form is a way of representing complex numbers using their magnitude and angle, while rectangular form uses the real and imaginary components. In polar form, a complex number is written as r(cosθ + isinθ), where r is the magnitude and θ is the angle in radians. In rectangular form, a complex number is written as a + bi, where a is the real component and b is the imaginary component.

How do you convert a complex number from polar form to rectangular form?

To convert from polar form to rectangular form, you can use the following formulas:
Real component: a = rcosθ
Imaginary component: b = rsinθ
For example, if a complex number is given in polar form as 3(cosπ/4 + isinπ/4), then the equivalent rectangular form would be 3(√2/2 + i√2/2).

How do you find the magnitude and angle of a complex number in polar form?

To find the magnitude (r) of a complex number in polar form, you can use the Pythagorean theorem: r = √(a^2 + b^2), where a and b are the real and imaginary components, respectively. To find the angle (θ), you can use the inverse tangent function: θ = tan^-1(b/a).
For example, if a complex number is given in polar form as 4(cosπ/3 + isinπ/3), then the magnitude is 4 and the angle is π/3 (or 60 degrees).

What is the principal argument of a complex number?

The principal argument of a complex number is the angle between the positive real axis and the line connecting the origin to the complex number in the complex plane. It is also known as the principal value of the argument and is typically represented by the symbol arg(z). The principal argument is always between -π and π radians.

How do you solve problems involving arg(z)?

To solve problems involving arg(z), you can use the properties of the argument function:
1. arg(z1z2) = arg(z1) + arg(z2)
2. arg(z1/z2) = arg(z1) - arg(z2)
3. arg(z^n) = narg(z)
You can also use the inverse tangent function to find the angle of a complex number in rectangular form, and then convert it to the principal argument by adding or subtracting multiples of 2π if necessary.

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