Product of Exponential Form (easy)

[DEMJ]

Homework Statement

$$e^{i\theta_1}e^{i\theta_2} = e^{i(\theta_1 + \theta_2)}}$$

Homework Equations

The Attempt at a Solution

For some reason every I multiply $$(cos\theta_1 + isin\theta_1)(cos\theta_2 + isin\theta_2)$$ I am getting

$$(cos\theta_1 cos\theta_2 + sin\theta_1 sin\theta_2) + i(sin\theta_1 cos\theta_2 + cos\theta_1 sin\theta_2)$$

according to my book the first part should be $$(cos\theta_1 cos\theta_2 - sin\theta_1 sin\theta_2)$$

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

What I am doing is $$cos\theta_1 cos\theta_2 - (isin\theta_1)(isin\theta_2) = cos\theta_1 cos\theta_2 - i^2 sin\theta_1 sin\theta_2$$ since $$i^2 = -1$$ 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]

$$e^{ix} = \cos x + i\sin x,\,\!$$

Substitute x=$\theta_1 + \theta_2$ and then use:

$$\sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \,$$

$$\cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta\,$$

 

[Mark44]

Homework Statement

$$e^{i\theta_1}e^{i\theta_2} = e^{i(\theta_1 + \theta_2)}}$$

Homework Equations

The Attempt at a Solution

For some reason every I multiply $$(cos\theta_1 + isin\theta_1)(cos\theta_2 + isin\theta_2)$$ I am getting

$$(cos\theta_1 cos\theta_2 + sin\theta_1 sin\theta_2) + i(sin\theta_1 cos\theta_2 + cos\theta_1 sin\theta_2)$$

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 $$(cos\theta_1 cos\theta_2 - sin\theta_1 sin\theta_2)$$

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

What I am doing is $$cos\theta_1 cos\theta_2 - (isin\theta_1)(isin\theta_2) = cos\theta_1 cos\theta_2 - i^2 sin\theta_1 sin\theta_2$$ since $$i^2 = -1$$ 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 $$(cos\theta_1 + isin\theta_1)(cos\theta_2 + isin\theta_2)$$

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

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