Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 1 / 15 / c

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- Solution c

Solution

Factor the leading term out of the Maclaurin series for the sine:

sin x = x (1 - \frac{x ^{2}}{6} + \frac{x ^{4}}{120} + O (x^{6})) .

The binomial theorem gives, through degree four inside the bracket,

(1 - \frac{x ^{2}}{6} + \frac{x ^{4}}{120} + O (x^{6}))^{6} = 1 + 6 (- \frac{x ^{2}}{6} + \frac{x ^{4}}{120}) + (2 6) \frac{x ^{4}}{36} + O (x^{6}) .

The coefficient of

x^{4}

is

1/20 + 5/12 = 7/15

. Multiplying back by

x^{6}

yields

sin^{6} x = x^{6} - x^{8} + \frac{7}{15} x^{10} + O (x^{12}) .

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