The function powsin(p) returns the formal power series that is equivalent to $\sin \left(p\right)$.

The powseries function evalpow accepts the form as any of powsin, Sin, or sin.

The command with(powseries,powsin) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{powseries}\right)\:$

$a≔\mathrm{powsin}\left(x^2\right)\:$

$b≔\mathrm{tpsform}\left(a\,x\,10\right)$

$b≔x^2-\frac{1}{6}{x}^6+O\left(x^{10}\right)$

$c≔\mathrm{evalpow}\left(\sin \left(\exp \left(x\right)\right)\right)\:$

$d≔\mathrm{tpsform}\left(c\,x\,5\right)$

$d≔\sin \left(1\right)+\cos \left(1\right)x+\left(-\frac{\sin \left(1\right)}{2}+\frac{\cos \left(1\right)}{2}\right)x^2-\frac{1}{2}\sin \left(1\right)x^3+\left(-\frac{5\cos \left(1\right)}{24}-\frac{\sin \left(1\right)}{4}\right)x^4+O\left(x^5\right)$

$e≔\mathrm{powdiff}\left(\mathrm{powsin}\left(x\right)\right)\:$

$f≔\mathrm{tpsform}\left(e\,x\,6\right)$

$f≔1-\frac{1}{2}{x}^2+\frac{1}{24}{x}^4+O\left(x^6\right)$