 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2003/paper-1/8/a/ii/solution

The identity

sin^{3} x = (3 sin x - sin 3 x) /4

makes the lower-order cancellation explicit. Expand the two sines through degree seven:

sin^{3} x = x^{3} - \frac{x ^{5}}{2} + \frac{13 x ^{7}}{120} + O (x^{9}) .

These are the first three nonzero terms; there is no linear term.

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