Discover the Real Part of i*sin(∏/4 + i)

[MissP.25_5]

Hello, everyone.

Can some help me finish this solution? I am stuck. The questions is to find the real part of
i*sin(∏/4 + i).

 

[LCKurtz]

Hello, everyone.

Can some help me finish this solution? I am stuck. The questions is to find the real part of
i*sin(∏/4 + i).

Use $e^{i\theta} = \cos \theta + i\sin\theta$ and its conjugate with $\theta = i$.

 

[MissP.25_5]

Use $e^{i\theta} = \cos \theta + i\sin\theta$ and its conjugate with $\theta = i$.

Which part should I use that? I don't get it.

 

[LCKurtz]

Solve the Euler equations for $e^{i\theta}$ and $e^{-i\theta}$ for the sine and cosine in terms of the exponentials. Surely your book has those formulas.

 

[MissP.25_5]

Solve the Euler equations for $e^{i\theta}$ and $e^{-i\theta}$ for the sine and cosine in terms of the exponentials. Surely your book has those formulas.

Does that mean I didn't have to use the trig identity?

 

[LCKurtz]

You just need $\sin i$ and $\cos i$ to finish, don't you? You can get them from the above formulas.

 

[MissP.25_5]

You just need $\sin i$ and $\cos i$ to finish, don't you? You can get them from the above formulas.

Err...you mean like this? But it gets complicated.

 

[MissP.25_5]

You just need $\sin i$ and $\cos i$ to finish, don't you? You can get them from the above formulas.

Wait, I think I got it. Hold on, let me try and solve it and I will show it you.

 

[MissP.25_5]

You just need $\sin i$ and $\cos i$ to finish, don't you? You can get them from the above formulas.

Is this correct?

 

[LCKurtz]

That looks correct. Notice that you could have substituted the exponentials for $\sin(\frac \pi 4 + i)$ directly, avoiding using the addition formulas.

 

[MissP.25_5]

That looks correct. Notice that you could have substituted the exponentials for $\sin(\frac \pi 4 + i)$ directly, avoiding using the addition formulas.

How to substitute the exponentials for $\sin(\frac \pi 4 + i)$ directly?

 

[LCKurtz]

$\sin \theta = \frac {e^{i\theta}-e^{-i\theta}}{2i}$, the same formula you used before.

 

[MissP.25_5]

$\sin \theta = \frac {e^{i\theta}-e^{-i\theta}}{2i}$, the same formula you used before.

You mean like this? But then how do I finish it? Looks complicated there.

 

[SammyS]

You mean like this? But then how do I finish it? Looks complicated there.

What happened to $\ i\$ in the denominator?

Do you know what $\ e^{i\pi/4}\$ is ?

 

[MissP.25_5]

What happened to $\ i\$ in the denominator?

Do you know what $\ e^{i\pi/4}\$ is ?

I forgot to write the i.
##\ e^{i\pi/4}\ is equals to cos∏/4 + isin∏/4, right? And that makes it equals to 1/√2 + i/√2, right?

 

[SammyS]

I forgot to write the i.
##\ e^{i\pi/4}\ is equals to cos∏/4 + isin∏/4, right? And that makes it equals to 1/√2 + i/√2, right?

Yes, and $\ e^{-i\pi/4}= \ ?$

 

[MissP.25_5]

Yes, and $\ e^{-i\pi/4}= \ ?$

Thanks, I got it!Yay! Thank you!

 

[SammyS]

Thanks, I got it!Yay! Thank you!

Good!