 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-1/12/a/iii/solution

sin (u + i v) = sin u cosh v + i cos u sinh v

follows from the exponential definition of sine. With

u = π /2

, its imaginary term vanishes, while

cosh (ln 2) = \frac{e ^{l n 2} + e ^{- l n 2}}{2} = \frac{2 + 1/2}{2} = \frac{5}{4} .

Consequently the real part is $5/4$ and the imaginary part is $0$ .

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