Product of Exponential Form (easy)

[DEMJ]

Homework Statement

[tex]e^{i\theta_1}e^{i\theta_2} = e^{i(\theta_1 + \theta_2)}}[/tex]

Homework Equations

The Attempt at a Solution

For some reason every I multiply [tex](cos\theta_1 + isin\theta_1)(cos\theta_2 + isin\theta_2)[/tex] I am getting

[tex](cos\theta_1 cos\theta_2 + sin\theta_1 sin\theta_2) + i(sin\theta_1 cos\theta_2 + cos\theta_1 sin\theta_2)[/tex]

according to my book the first part should be [tex](cos\theta_1 cos\theta_2 - sin\theta_1 sin\theta_2)[/tex]

what am I missing here? Is it some basic fundamental from calculus I have forgotten?

What I am doing is [tex]cos\theta_1 cos\theta_2 - (isin\theta_1)(isin\theta_2) = cos\theta_1 cos\theta_2 - i^2 sin\theta_1 sin\theta_2[/tex] since [tex]i^2 = -1[/tex] that makes it positive. But this can't be right because both the book and my notes from class cannot be wrong. So please enlighten me =]

 

[njama]

[tex]e^{ix} = \cos x + i\sin x,\,\![/tex]

Substitute x=[itex]\theta_1 + \theta_2[/itex] and then use:

[tex]\sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \,[/tex]

[tex]\cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta\,[/tex]

 

[Mark44]

Homework Statement

[tex]e^{i\theta_1}e^{i\theta_2} = e^{i(\theta_1 + \theta_2)}}[/tex]

Homework Equations

The Attempt at a Solution

For some reason every I multiply [tex](cos\theta_1 + isin\theta_1)(cos\theta_2 + isin\theta_2)[/tex] I am getting

[tex](cos\theta_1 cos\theta_2 + sin\theta_1 sin\theta_2) + i(sin\theta_1 cos\theta_2 + cos\theta_1 sin\theta_2)[/tex]

For the real part you should be getting cos(th1)cos(th2) + i^2*sin(th1)sin(th2). I think you omitted the i^2 factor.

according to my book the first part should be [tex](cos\theta_1 cos\theta_2 - sin\theta_1 sin\theta_2)[/tex]

what am I missing here? Is it some basic fundamental from calculus I have forgotten?

What I am doing is [tex]cos\theta_1 cos\theta_2 - (isin\theta_1)(isin\theta_2) = cos\theta_1 cos\theta_2 - i^2 sin\theta_1 sin\theta_2[/tex] since [tex]i^2 = -1[/tex] that makes it positive. But this can't be right because both the book and my notes from class cannot be wrong. So please enlighten me =]

 

[Subdot]

For some reason every I multiply [tex](cos\theta_1 + isin\theta_1)(cos\theta_2 + isin\theta_2)[/tex]

...What I am doing is [tex]cos\theta_1 cos\theta_2 - (isin\theta_1)(isin\theta_2) = cos\theta_1 cos\theta_2 - i^2 sin\theta_1 sin\theta_2[/tex] since [tex]i^2 = -1[/tex] that makes it positive.

Why are you subtracting? Once you answer that, you should be all set.