Real and imaginary values of complex numbers

For each complex number z, find Re z (real z), Im z (imaginary z), and magnitude of z.


1. z = e^(1-2i)
2. z = i^(1/2)
3. z = i^i
4. z = tan(i)
5. z = [(1-i)/(2^(1/2))]^40

I'm not really sure how to even approach these so any help would be appreciated. And if possible please include steps so I can learn how to do these types of problems

1) z = e^(1 - 2i)
= e * e^(-2i)
= e * (cos(-2) + i sin(-2)), via e^(it) = cos t + i sin t
= e (cos 2 - i sin 2)

So, Re z = e cos 2, Im z = e sin 2, |z| = e √(cos^2(2) + sin^2(2)) = e.
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2) z = i^(1/2)
= (e^(πi/2))^(1/2)
= (e^(πi/2 + 2πik))^(1/2) for any integer k
= e^(πi/4 + πik), k = 0 or 1
= e^(πi/4) or e^(5πi/4)
= ± (1/√2 + i/√2); there are two values.

**If you only need principal value, ignore the negative answers.
Re z = ± 1/√2, Im z = ± 1/√2, |z| = √(1/2 + 1/2) = 1.
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3) z = i^i
= (e^(πi/2))^i
= (e^(πi/2 + 2πik))^i for any integer k
= e^(-π/4 - πk), for any integer k

**For principal value, take k = 0.

Re z = e^(-π/4 - πk), Im z = 0, |z| = e^(-π/4 - πk).
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4) z = tan i
= sin i / cos i
= [(1/(2i)) (e^(i * i) - e^(-i * i))] / [(1/2) (e^(i * i) + e^(-i * i))]
= -i (e^(-1) - e) / (e^(-1) + e)
= i (e - e^(-1)) / (e + e^(-1)).

Re z = 0, Im z = (e - e^(-1)) / (e + e^(-1)), |z| = (e - e^(-1)) / (e + e^(-1)).
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5) z = [(1 - i)/√2]^40
= [e^(-πi/4)]^40
= e^(-10πi)
= cos(-10π) + i sin(-10π)
= 1.

So, Re z = 1, Im z = 0, |z| = 1.

I hope this helps!