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is there an identity like sinjx = sinhx or something similar?
is there an identity like sinjx = sinhx or something similar? To convert between the two using imaginary numbers.
3 Answers
- Alam Ko IyanLv 71 decade agoFavorite Answer
sinh x = [ e^x - e^(-x) ]/2
i sin w = [ e^(iw) - e^(-iw) ]/2
to relate: x = iw or w = -ix
Thus
sinh x = i sin(-ix) = -i sin(ix)
d:
- Anonymous1 decade ago
sinjx = sinhx when j = h
- Anonymous1 decade ago
Yes.
e^ix = cos(x) + isin(x) (1)
e^-ix = cos(x) - isin(x) (2)
so take (1)-(2) to get rid of cos(x)
[cos(x) + isin(x)] - [cos(x) - isin(x)] = e^ix - e^-ix
isin(x) = 1/2*(e^ix - e^-ix) = sinh(ix)
isin(x) = sinh(ix)
replace x with -i*y to get
isin(-iy) = sinh(i*-i*y)
-isin(y) = sinh(y)
you can use this method to also prove similar identities for cos(ix) and cosh(ix)
