# Evaluate the complex number.

• joelstacey
If you have a complex number ##z = a + ib##, how do you find the modulus?The modulus is found by taking the length of the vector in polar coordinates that is parallel to the real axis and the imaginary axis, and squaring it.f

#### joelstacey

Homework Statement
For each expression, evaluate its real part, imaginary part, modulus and argument.
1. (e^(i*theta))^2
Relevant Equations
r*e^(i*theta)= r*(cos(theta) + i*sin(theta))
(e^(i*theta))^2 = (sin(theta)+i*cos(theta))^2 = cos(theta)^2 - sin(theta)^2 + 2*i*sin(theta)*cos(theta), so the real part would be: cos(theta)^2 - sin(theta)^2, and the imaginary part would be: 2*i*sin(theta)*cos(theta). But then I don't know where to start with the modulus or the argument?

What are the definition of the modulus and argument of a complex number?

What are the definition of the modulus and argument of a complex number?
I know the modulus is the length of the vector in an argand diagram, and the modulus is the angle it makes with the x axis, but since it is squared i don't see how it works as an argand diagram.

You can also write ##(e^{i\theta})^2## as ##e^{2i\theta}##, using the properties of exponents.
Rewriting the above using Euler's formula, we have ##e^{2i\theta} = \cos(2\theta) + i\sin(2\theta)##, which agrees with what you found for the real and imaginary parts (after using double angle identities).

If you have a complex number ##z = a + ib##, how do you find the modulus? From the above, the argument (arg) should be simple to find.

I guess a couple of things
1.) Given the real and complex parts, you could write down a new polar coordinates form. The modulus is not hard to compute this way, though I will admit the argument requires knowing some trig trickery.
2.) There's a much simpler way to do this.
## (e^{i\theta})^2 = e^{i\theta} e^{i\theta} = e^{i\theta+i\theta}##.

Note ##(e^a)^b \neq e^{ab}## in general, but when b is an integer you can do this.

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so the real part would be: cos(theta)^2 - sin(theta)^2, and the imaginary part would be: 2*i*sin(theta)*cos(theta). But then I don't know where to start with the modulus or the argument?
Do you know the trig identities for ##\cos(2\theta)## and ##\sin(2\theta) ## ?