Let x=(2^-1) and y=(12^-143).

Question

Let $x=\sin(2	an^{-1}\alpha)$ and $y=\sin\left(\dfrac12	an^{-1}\dfrac43ight)$. If $S=\{\alpha\in\mathbb{R}: y^2=1-x\}$, then $\displaystyle\sum_{\alpha\in S}16\alpha^3$ is equal to

Answer: 130

Accepted Answer

$x=\sin(2\tan^{-1}\alpha)=\dfrac{2\alpha}{1+\alpha^2}$. Also, $\dfrac12\tan^{-1}\dfrac43=\sin^{-1}\dfrac{1}{\sqrt5}$, so $y=\dfrac{1}{\sqrt5}$ and $y^2=\dfrac15$.

$$\frac15=1-\frac{2\alpha}{1+\alpha^2}\ \Rightarrow\ 1+\alpha^2=5+5\alpha^2-10\alpha\ \Rightarrow\ 2\alpha^2-5\alpha+2=0\ \Rightarrow\ \alpha=2,\ \frac12.$$

$$\sum_{\alpha\in S}16\alpha^3=16(2)^3+16\left(\frac12\right)^3=128+2=130.$$

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