  We consider the $\alpha$-sine transform of the form $T_\alpha
f(y)=\int_0^\infty\vert\sin(xy)\vert^\alpha f(x)dx$ for $\alpha>-1$, where $f$
is an integrable function on $\mathbb{R}_+$. First, the inversion of this
transform for $\alpha>1$ is discussed in the context of a more general family
of integral transforms on the space of weighted, square-integrable functions on
the positive real line. In an alternative approach, we show that the
$\alpha$-sine transform of a function $f$ admits a series representation for
all $\alpha>-1$, which involves the Fourier transform of $f$ and coefficients
which can all be explicitly computed with the Gauss hypergeometric theorem.
Based on this series representation we construct a system of linear equations
whose solution is an approximation of the Fourier transform of $f$ at
equidistant points. Sampling theory and Fourier inversion allow us to compute
an estimate of $f$ from its $\alpha$-sine transform. The same approach can be
extended to a similar $\alpha$-cosine transform on $\mathbb{R}_+$ for
$\alpha>-1$, and the two-dimensional spherical $\alpha$-sine and cosine
transforms for $\alpha>-1$, $\alpha\neq 0,2,4,\dots$. In an extensive numerical
analysis, we consider a number of examples, and compare the inversion results
of both methods presented.
