When we know the cos adj/hyp to find an angle why do we use the inverse of cos to find theta?
You don't use cos = adj/hyp to find an angle, you use cos = adj/hyp to get the cosine of the angle, which is a number. If you want to find the angle whose cosine is equal to a given number, you use the inverse cosine.
Just as we use the inverse of the exponential to solve e^x = c, getting ln(e^x) = ln(c) so that x = ln(c), we generally use the inverse function to solve any equation of the form f(x) = c, where x is the variable and c is a known value.
The symbolic representation of this process is
f^-1 ( f(x) ) = (f^-1) (c) so that
x = (f^-1) (c).
Here f^-1 is understood to be the inverse function. For example if f(x) = e^x, then f^-1(x) = ln(x).
Note that the inverse function has nothing whatsoever to do with the reciprocal of the function.
The cosine function applies to angles and for any angle it gives you a number between -1 and 1.
The inverse cosine function applies to numbers between -1 and 1 and gives you an angle between 0 and pi.
You use the cosine to get numbers from angles, and the inverse cosine to get angles from numbers.