 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2020/paper-1/11/a/ii/solution

Powers of the imaginary unit repeat in blocks

i, - 1, - i, 1

, each with sum zero. The first

44

terms comprise eleven complete blocks; the last is

i^{45} = i

. Applying Euler's formula to the resulting complex exponential gives

e^{i} = cos 1 + i sin 1

. Hence

ℜ z = cos 1, ℑ z = sin 1.

The angles here are in radians.

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