 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2016/2/solution

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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