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

  • Thread starter MissP.25_5
  • Start date
In summary, you can substitute the exponentials for ##\sin(\frac \pi 4 + i)## directly to solve for the sine and cosine.
  • #1
MissP.25_5
331
0
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).
 

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  • #2
MissP.25_5 said:
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##.
 
  • #3
LCKurtz said:
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.
 
  • #4
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.
 
  • #5
LCKurtz said:
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?
 
  • #6
You just need ##\sin i## and ##\cos i## to finish, don't you? You can get them from the above formulas.
 
  • #7
LCKurtz said:
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.
 

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  • #8
LCKurtz said:
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.
 
  • #9
LCKurtz said:
You just need ##\sin i## and ##\cos i## to finish, don't you? You can get them from the above formulas.

Is this correct?
 

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  • #10
That looks correct. Notice that you could have substituted the exponentials for ##\sin(\frac \pi 4 + i)## directly, avoiding using the addition formulas.
 
  • #11
LCKurtz said:
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?
 
  • #12
##\sin \theta = \frac {e^{i\theta}-e^{-i\theta}}{2i}##, the same formula you used before.
 
  • #13
LCKurtz said:
##\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.
 

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  • #14
MissP.25_5 said:
You mean like this? But then how do I finish it? Looks complicated there.
attachment.php?attachmentid=70243&d=1401677421.jpg


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

Do you know what ##\ e^{i\pi/4}\ ## is ?
 
  • #15
SammyS said:
attachment.php?attachmentid=70243&d=1401677421.jpg


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?
 
  • #16
MissP.25_5 said:
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}= \ ? ##
 
  • #17
SammyS said:
Yes, and ##\ e^{-i\pi/4}= \ ? ##

Thanks, I got it!Yay! Thank you!
 
Last edited:
  • #18
MissP.25_5 said:
Thanks, I got it!Yay! Thank you!

Good!
 

1. What is the real part of i*sin(∏/4 + i)?

The real part of i*sin(∏/4 + i) is 0.707, also known as the square root of 2 divided by 2.

2. How is the real part of i*sin(∏/4 + i) calculated?

The real part of i*sin(∏/4 + i) is calculated by taking the sine of ∏/4, which is equal to 0.707, and then multiplying it by the imaginary number i. This results in a complex number with a real part of 0.707 and an imaginary part of 0.707i.

3. What does i*sin(∏/4 + i) represent?

i*sin(∏/4 + i) represents a complex number with both a real and imaginary component. It is used to solve problems in mathematics and physics that involve complex numbers.

4. Can the real part of i*sin(∏/4 + i) be negative?

No, the real part of i*sin(∏/4 + i) cannot be negative because the sine of ∏/4 is always positive, and multiplying it by i does not change its sign.

5. How is the real part of i*sin(∏/4 + i) useful in real-world applications?

The real part of i*sin(∏/4 + i) is useful in real-world applications such as signal processing and electrical engineering, where complex numbers are used to represent frequency and amplitude. It is also used in quantum mechanics to describe the wave function of particles.

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