Lets say for example I have a triangle like the one below:
If I find x by doing
$\sin^{-1}\dfrac {3}{5}$
is the resulting number in degrees or radians? I understand that a calculator can be set to return degrees or radians, but what is this number inherently? If I have misunderstood something then please let me know.
This mnemonic isn't inherently in any specific unit for measuring angles. The sine is $3/5$; that's not a convention of units, just an objective fact. Now, if you run it through an inverse sine calculator, it'll return an answer in whatever units it's designed to. Calculators typically have degree, radian and grad conventions you can switch between. And that convention is used when interpreting what angle you'd like the sine of, if you ask for it. If $\sin 30$ comes out as $0.5$, you're working in degrees. But that's all it is, a convention.