# Find best element of continuous approximation for the f ( x ) = sin &#x2061;<!-- ⁡ -

Find best element of continuous approximation for the
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Alexzander Bowman
If you want the "best" approximation of $\mathrm{sin}\left(x\right)$ for ${x}_{1}\le x\le {x}_{2}$ using (say) a cubic polynomial, you could minimize with respect to $\left(a,b\right)$
$I={\int }_{{x}_{1}}^{{x}_{2}}{\left(\mathrm{sin}\left(x\right)-\left(ax+b{x}^{3}\right)\right)}^{2}$
This would correspond to a linear regression with an infinite number of data points.
Using ${x}_{1}=0$ and ${x}_{2}=\frac{\pi }{4}$, this would lead to
$a=\frac{80640\sqrt{2}-20160\sqrt{2}\pi -1920\sqrt{2}{\pi }^{2}+60\sqrt{2}{\pi }^{3}}{{\pi }^{5}}\approx 0.999259$
$b=-\frac{2150400\sqrt{2}-537600\sqrt{2}\pi -53760\sqrt{2}{\pi }^{2}+2240\sqrt{2}{\pi }^{3}}{{\pi }^{7}}\approx -0.161035$
For these values $I\approx 6.81×{10}^{-9}$ while using Taylor coefficients $\left(a=1,b=-\frac{1}{6}\right)$ we should get $I\approx 432×{10}^{-7}$ that is to say almost $64$ times larger.