Homework Help: Addition of Cis(x)

You're going to have a hard time proving that, unless my math is bad today (a very real possibility... I've been making some pretty bad gaffes lately...)

I get that it should come out to 2isin(x1-x2).

 

Use the addition formulas for trigonometric functions:

cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

sin(A + B) = sin(A)cos(B) + sin(B)cos(A)

EDIT: AUMathTutor is right.

 

dx, do you get the answer the OP is looking for when you work it out that way?

I used the rules cos(x) = cos(-x) and sin(x) = -sin(-x) and got it my way.

 

In fact,

cis(x1 - x2) - cis(x2 - x1) = 2cos(x1 - x2)

Is clearly wrong. try x1 = x2 = 0. You get

cis(0) - cis(0) = 2cos(0) = 2

so 0 = 2. I think I'm alright on this one today.

 

You're right AUMathTutor, the answer is 2isin(x₁-x₂).

 

Here'w what you need to do:

1. Write down the problem in terms of cis.
2. Rewrite the problem in terms of sin and cos using the definition of cis.
3. Apply the equalities sin(x) = -sin(-x) and cos(x) = cos(-x).
4. Collect like terms and/or cancel out terms.
5. See whether you get what you wanted.

The problem is incorrect. The answer is 2isin(x1-x2). You'll see the cosines cancel out. This is just some simple algebra.

You can also use the rules dx posted, and then recombine your answer to get the same thing. It's a few more lines of math, but probably a little more clear.

 

It can also be seen very easily by drawing them in the complex plane, if you've been taught that.

 

[itex]cis(\theta)[/itex] is the point where the unit circle crosses the line through (0,0) that makes angle [itex]\theta[/itex] with the positive x-axis. Surely that is not hard to find on a plane.

 

"The second equation is true only when x1 - x2 = 0."

Any reasonable person can see he meant to write x2 - x1.