 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-1/11/d/solution

sin 5 θ

(cos θ + i sin θ)^{5}

. The binomial theorem gives, with

s = sin θ

sin 5 θ = 5 cos^{4} θ s - 10 cos^{2} θ s^{3} + s^{5} .

Use

cos^{2} θ = 1 - s^{2}

and simplify:

sin 5 θ = 16 s^{5} - 20 s^{3} + 5 s .

The requested equation cancels the

16 s^{5}

term, leaving

5 s (1 - 4 s^{2}) = 0

. Thus

s = 0

s = \pm 1/2

. The complete list in

0 \leq θ < 2 π

θ = 0, \frac{π}{6}, \frac{5 π}{6}, π, \frac{7 π}{6}, \frac{11 π}{6} .

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