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.??
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.
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).
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).
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.
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.