Past exam of the mathematical methods I course of the University of Oxford / 2016 / 2

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- Solution 2

Solution

The complex sine identity is

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

It follows either from the exponential definition of sine or its addition formula, using

sin (i y) = i sinh y

and

cos (i y) = cosh y

. Consequently

Re sin (2 + 3 i) = sin 2 cosh 3, Im sin (2 + 3 i) = cos 2 sinh 3.

2

and

3

in these functions are dimensionless; the real trigonometric argument is measured in radians.

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