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

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- Solution c

Solution

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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