Solve sum_x=1^15left(sin((2x-1)2)right)

Evaluate

\sin(2)+\sin(6)+\sin(10)+\sin(14)+\sin(18)+\sin(22)+\sin(26)+\sin(30)+\sin(34)+\sin(38)+\sin(42)+\sin(46)+\sin(50)+\sin(54)+\sin(58)\approx 1.073583254

Quiz

Trigonometry

\sum_{ x=1 }^{ 15 } \left( \sin ( (2x-1)2 ) \right)

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As an alternative approach, you may notice that \begin{eqnarray*} S = \frac{1}{10}\sum_{n\geq 1}(-1)^n\left(\frac{1}{6n-5}-\frac{1}{6n+5}\right)&=&\frac{1}{10}\sum_{n\geq 1}(-1)^n\int_{0}^{1}\left(x^{6n-6}-x^{6n+4}\right)\,dx\\&=&\frac{1}{10}\int_{0}^{1}\frac{x^{10}-1}{1+x^6}\,dx\\&=&\frac{1}{10}\left(\frac{1}{5}-\int_{0}^{1}\frac{1+x^4}{1+x^6}\,dx\right)\\&=&\frac{1}{10}\left(\frac{1}{5}-\int_{0}^{1}\frac{1+x^4}{(1+x^2)(1-x^2+x^4)}\,dx\right)\\ &=&\frac{1}{10}\left(\frac{1}{5}-\left[\frac{2}{3}\arctan(x)+\frac{1}{3}\arctan\frac{x}{x^2-1}\right]_{0}^{1}\right)\\&=&\frac{1}{10}\left(\frac{1}{5}-\frac{\pi}{3}\right)=\color{red}{\frac{1}{50}-\frac{\pi}{30}}.\end{eqnarray*}

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