sin-1(1.04)

There are no real solutions to this.

$$-1\le sin\theta \le 1$$

sin-1(1.04) ?

If 1.04 is a complex number z:

$$\\ \small{\text{
$
\boxed{
\sin^{-1}(z) = -i\cdot \log\left(\sqrt{1-z^2}+i\cdot z \right)
\qquad $Complex logarithm $\log{(z)} = \ln{(|z|)}+i\cdot \varphi
}
$
}}\\\\
\small{\text{We have $z = 1.04$}}\\
\small{\text{
\sin^{-1}(z)=sin^{-1}(1.04) = -i\cdot \log\left(\sqrt{1-(1.04)^2}+i\cdot 1.04 \right)
$}}\\
\small{\text{
sin^{-1}(1.04) = -i\cdot \log\left(\sqrt{1-(1.04)^2}+i\cdot 1.04 \right)
$}}\\
\small{\text{
sin^{-1}(1.04) = -i\cdot \log\left(\sqrt{ -0.0816 }+i\cdot 1.04 \right)
$}}\\
\small{\text{
sin^{-1}(1.04) = -i\cdot \log\left(\sqrt{0.0816}\cdot \sqrt{-1}+i\cdot 1.04 \right)
$}}\\
\small{\text{
sin^{-1}(1.04) = -i\cdot \log\left(\sqrt{0.0816}\cdot i+i\cdot 1.04 \right)
$}}\\
\small{\text{
sin^{-1}(1.04) = -i\cdot \log\left(0.28565713714\cdot i+i\cdot 1.04 \right)
$}}\\
\small{\text{
sin^{-1}(1.04) = -i\cdot \log\left(0.28565713714\cdot i+i\cdot 1.04 \right)
$}}\\
\small{\text{
sin^{-1}(1.04) = -i\cdot \mathbf{
\log{\left( 1.32565713714 \cdot i \right)} }
$}}\\\\
\small{\text{
a+b\cdot i = \mathbf{1.32565713714 \cdot i } \qquad a= 0 $ and $ b = 1.32565713714
$}}\\
\small{\text{
$\textcolor[rgb]{0,0,1}{|1.32565713714 \cdot i|} = \sqrt{a^2+b^2} = \sqrt{0^2+1.32565713714^2} = \textcolor[rgb]{0,0,1}{1.32565713714}
$}}\\
\small{\text{
$\varphi = \tan^{-1}{ \left(\dfrac{1.32565713714}{0} \right) }
\quad \Rightarrow \quad \varphi = \textcolor[rgb]{1,0,0}{\dfrac{\pi}{2}}
$}} \\\\
\small{\text{
$\log{\left( 1.32565713714 \cdot i \right)}
= \ln{ ( \textcolor[rgb]{0,0,1}{ 1.32565713714 } ) } + i \cdot \textcolor[rgb]{1,0,0}{ \dfrac{\pi}{2} }
$}}\\\\
\small{\text{
sin^{-1}(1.04) = -i\cdot \left(
\ln{ ( 1.32565713714 ) } + i \cdot \dfrac{\pi}{2}
\right) \quad \ln{ ( 1.32565713714 ) } = 0.28190828905
$}}\\
\small{\text{
sin^{-1}(1.04) = -i\cdot \left(
0.28190828905 + i \cdot \dfrac{\pi}{2}
\right)
$}}\\\\
\small{\text{
sin^{-1}(1.04) = -i\cdot
0.28190828905 - i^2 \cdot \dfrac{\pi}{2}
\right) \quad | \quad \boxed{i^2=-1}
$}}$$

$$\\ \small{\text{
$
sin^{-1}(1.04) = \dfrac{\pi}{2} - 0.28190828905 \cdot i
$}}\\
\small{\text{
$
sin^{-1}(1.04) = 1.57079632679 - 0.28190828905 \cdot i
$}}\\$$

sin-1(1.04) =1.0816