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Homework 5 (MATH 1199 Fall 2019)
1. Compute...
(a) 21−i
(b) (P.V.)(1+i)2i
(c) 1i
(d) (P.V.)(−2)−i
(e) arcsin(−i)
(f) Arccos(12)
2. Use the definitions of sin, cos, sinh, and cosh to show...
(a) ddzcos(z)=−sin(z)
(b) (double angle identity) sin(2z)=2sin(z)cos(z)
(c) ddzsinh(z)=cosh(z)
(d) cosh(iz)=cos(z)
3. Define tan(z)=sin(z)cos(z).
(a) Show that tan(z)=1i(eiz−e−izeiz+e−iz).
(b) Resembling p.39-40 in the notes, let z=tan(w) so that w=arctan(z). Express arctan(z) in terms of the logarithm. Hint: it should again become a quadratic in the variable v=eiw (hence: 1v=e−iw), but it should not require quadratic formula to solve.