find the modulus and the argument of a complex number with exponential

[Lolu12]

Hi, I have an exercise that asks me to find the argument and modulus of a complex number from the addition of 2 exponential, and I would need your help because I've been blocked for a long time, thank you for your help

 

[Harry_the_cat]

Can you write each term another way? And add them in the alternative form? Then find arg and mod of sum?

 

[Lolu12]

If i convert everything to cos and sin and that I factor the result, i get to that, but after i get stuck..
And thank you for your fast help !

 

[Lolu12]

I made an error on factorization, so here is the result without factorization:
cos(x)+isin(x)*(cos^2(x)-sin^2(x)+i*2*sin(x)*cos(x))
Sorry for the double step, but I couldn't find a way to modify the old one

 

[Deleted member 4993]

I made an error on factorization, so here is the result without factorization:
cos(x)+isin(x)*(cos^2(x)-sin^2(x)+i*2*sin(x)*cos(x))
Sorry for the double step, but I couldn't find a way to modify the old one

You write:

cos(x)+isin(x)*(cos^2(x)-sin^2(x)+i*2*sin(x)*cos(x)) = e^ix + e^2ix

However, it should be:

cos(x) + i * sin(x) + [cos^2(x) - sin^2(x) + i * 2 * sin(x) * cos(x)]

Now collect the real and imaginary terms and put those together. There is no need for factorization.

 

[Dr.Peterson]

This may be a lot easier to attack geometrically. What would this sum look like on the complex plane?

 

[pka]

Hi, I have an exercise that asks me to find the argument and modulus of a complex number from the addition of 2 exponential, and I would need your help because I've been blocked for a long time, thank you for your help

\(\exp(ix)=\cos(x)+i\sin(x)\) and
\(\exp(i2x)=\cos(2x)+i\sin(2x)\)

 

[Lolu12]

First, thank you for all the math leads to follow !
I gathered the terms, can I just give the imaginary and real part on this shape or do I have to simplify?
I would also like to know how to solve it geometrically, but I don't know what it will look like on a complex plane

 

[Dr.Peterson]

First, thank you for all the math leads to follow !
I gathered the terms, can I just give the imaginary and real part on this shape or do I have to simplify?
I would also like to know how to solve it geometrically, but I don't know what it will look like on a complex plane

You were asked for the argument and modulus, so if you continue as you are, you have to use the real and imaginary parts you have to find those.

One alternative is to factor out [MATH]e^{ix}[/MATH] from the two terms at the start, which may make that work easier.

For the geometrical approach, do you not know how to graph a complex number on the plane? The number a+ib is represented by the point (a,b); or you can think of it as the vector <a,b> if you are familiar with that concept. The important thing is to know what addition looks like.

It will be helpful if you tell us what you have learned. I expected that if you are studying complex exponentials, you must know about the graphical representation of them.

 

[pka]

Here is the square of the real part. Here is the square of the imaginary part.
Can you combine and find the square root to get the modulus?

 

[Lolu12]

By factoring from the start by an exponential, I have this result on the image, but i'm not sure it's good

I also know how to represent a complex on a plane, but I have never done it with cos sin or exponentials, always with integers

And this is what I found after calculating the modulus : sqrt(2)*sqrt(cos(x)+1)

 

[Dr.Peterson]

By factoring from the start by an exponential, I have this result on the image, but i'm not sure it's good

I also know how to represent a complex on a plane, but I have never done it with cos sin or exponentials, always with integers

And this is what I found after calculating the modulus : sqrt(2)*sqrt(cos(x)+1)

I don't know how you got your results; you'll have to show your work. But the modulus is correct.

I suggested factoring out [MATH]e^{ix}[/MATH] from [MATH]e^{ix} + e^{2ix}[/MATH], which gives you [MATH]e^{ix}(1 + e^{ix})[/MATH].

Since the first factor has modulus 1, the modulus is just the modulus of [MATH]1 + e^{ix}[/MATH], which is not hard to find.

Since this is a product, the argument is the sum of the arguments of the factors, which also are not hard to find.

I can't imagine teaching about the exponential form without ever showing how to plot it, using the modulus and argument, and how multiplication affects each.