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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).
LCKurtz said:Use ##e^{i\theta} = \cos \theta + i\sin\theta## and its conjugate with ##\theta = i##.
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.
LCKurtz said:You just need ##\sin i## and ##\cos i## to finish, don't you? You can get them from the above formulas.
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.
MissP.25_5 said:You mean like this? But then how do I finish it? Looks complicated there.
SammyS said:
What happened to ##\ i\ ## in the denominator?
Do you know what ##\ e^{i\pi/4}\ ## is ?
Yes, and ##\ e^{-i\pi/4}= \ ? ##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?
SammyS said:Yes, and ##\ e^{-i\pi/4}= \ ? ##
MissP.25_5 said:Thanks, I got it!Yay! Thank you!
The real part of i*sin(∏/4 + i) is 0.707, also known as the square root of 2 divided by 2.
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.
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.
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.
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.