 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2006/paper-1/3/c/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

Use the complex exponential forms of the sine and cosine, and put

w = e^{2 i z}

. Multiplying the equation by

2 e^{i z}

gives

(2 - i) (w - 1) / i + (w + 1) = 0

, which simplifies to

w = 1 - i

. Taking all branches of the complex logarithm,

2 i z = \frac{1}{2} ln 2 + i (- \frac{π}{4} + 2 πn) .

Thus every solution is

z = nπ - \frac{π}{8} - \frac{i}{4} ln 2, n \in Z .

The exponential factor used to multiply the equation is nonzero, so no solutions were introduced or lost.

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