- #1

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## Homework Statement

Prove that

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

## The Attempt at a Solution

Cis is a sin-and-cos summation. Shortly,

cisx = cosx + isinx

How can you prove the statement?

- Thread starter soopo
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- #1

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Prove that

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

Cis is a sin-and-cos summation. Shortly,

cisx = cosx + isinx

How can you prove the statement?

- #2

- 492

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

- #3

dx

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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.

- #4

- 492

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

- #5

- 492

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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.

- #6

dx

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You're right AUMathTutor, the answer is 2isin(x_{1}-x_{2}).

- #7

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Could you please, write your methods down.

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

The following must be true to prove the statement

cos(x1 - x2) = cos(x2 - x1)

and

sin(x1 - x2) = -sin(x1 - x2)

If the above statements hold, then the original statement can be proven true.

- #8

- 492

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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.

- #9

dx

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

- #10

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

I can now draw easily the result to a complex plane. I can also draw the result on LHS in the first post.

This shows me that the initial statement must be false.

However, I am not use how you can use complex plane without expanding cis parts.

I personally need to see the imag and real parts to draw the results on the plane.

- #11

HallsofIvy

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- #12

Mark44

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The two statements do not hold for all values of x1 and x2.Could you please, write your methods down.

The following must be true to prove the statement

cos(x1 - x2) = cos(x2 - x1)

and

sin(x1 - x2) = -sin(x1 - x2)

If the above statements hold, then the original statement can be proven true.

The first equation is identically true, because cos(x) = cos(-x) for all x.

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

- #13

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

- #14

Mark44

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- #15

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@Mark: Thank you for your correction!

I tried to get the above result by "brute force" without considering the situation in the unit circle.

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