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> if $a,b,c,d$ be the first positive solutions of $\sin x=\frac{1}{4}$ then find $$\sin (d/2)+2\sin (c/2)+3\sin (b/2)+4\sin (a/2)$$.$(a<b<c<d)$
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**Attempts**
$$\sin x=1/4$$,$$4\sin^2(x/2)(1-\sin^2(x/2))=\frac{1}{16}$$
Let $\sin^2 (x/2)=t$:it remains to solve $$t-t^2=\frac{1}{64}$$ .The rest is just brute force i.e we calculate the 4 values $\sin(x/2)$ can take , make sure the angles are arranged in ascending order and substitute the roots.As far as i can see i a getting irrational roots.

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My question is does anyone spot a trick, that can avoid all this work.!