 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-1/6/solution

Put

u = sin θ

and move all terms to one side. The polynomial factors as

u^{3} - u^{2} - \frac{u}{4} + \frac{1}{4} = (u - 1) (u - \frac{1}{2}) (u + \frac{1}{2}) .

u = 1

yields the already supplied

θ = π /2

. In the stated interval,

u = 1/2

gives

π /6, 5 π /6

, while

u = - 1/2

gives

- π /6, - 5 π /6

. Thus all the other solutions are

θ = - \frac{5 π}{6}, - \frac{π}{6}, \frac{π}{6}, \frac{5 π}{6} .

There are no endpoint solutions because

sin (\pm π) = 0

is not one of the three roots.

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