 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/ia/paper-1/5c/ii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

Using

cos z = (e^{i z} + e^{- i z}) /2

and collecting real and imaginary terms gives the complex cosine identity

cos (x + i y) = cos x cosh y - i sin x sinh y

. Thus

Re (cos z) = cos x cosh y, Im (cos z) = - sin x sinh y .

The sign follows from the factors

e^{- y}

and

e^{y}

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