 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-1/15/b/i/solution

Write the sine argument as

π /2 + δ

, with

δ = \frac{π}{2} (e^{x} - 1) = \frac{π}{2} x + \frac{π}{4} x^{2} + O (x^{3})

sin (π /2 + δ) = cos δ

1 - δ^{2} /2 + O (δ^{4})

, so

sin (\frac{π e ^{x}}{2}) = 1 - \frac{π ^{2}}{8} x^{2} + O (x^{3}) .

There is no linear term because the outer sine has zero derivative at

π /2

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