Does this calculator have an inverse function? If so, which button is it?

I suspect that you are confusing things a bit.
The inverse sine, called the arcsine is a function defined in the closed interval [-1,1]. And so is the inverse cosine. Any value outside this interval will give you a non-real result (meaning a complex one).
There are no limitations on the domain of definition of the inverse hyperbolic sine or sinh^-1
If your input value is allowed to be complex, the arcsine function gives a complex value. See the screen capture

For the reciprocal of a number, enter the number then press the key marked with an x^-1. You might have to use the SHIFT key first. For inverse of trigonometric functions use the SHIFT key followed by the function name. The inverse of LN is e^x. The inverse of log is 10^x. To calculate the inverse of a matrix, enter the name of the (already created) square matrix then press x^-1.

By 'log inverse', you presumably mean the inverse of the logarithm function. There are two logarithm functions on most scientific calculators. Firstly [ln], or natural logarithms, to the base e, where the inverse is e^x. Secondly [log], or logarithms to the base 10, where the inverse is 10^x.

Example 1 : ln(2) = 0.69314718 so e^0.69314718 = 2
Example 2: log(2) = 0.301029995 so 10^0.30102995 = 2

The inverse of what&? Each inverse function uses the same key (shifted) as its direct function. For the reciprocal key look for x^-1 or [1/X]

The inverse of the log function is the power function.
For log in base 10 that inverse is 10 to a power of
More generally, let b be the base of the logarithm. If y=log_b (x) then x=b^y
For your case log=log_10, to calculate the inverse you perform 10^(-2)=0.01=1/100
On calculators the log in base 10 and its inverse share the same physical key. One is accessed directly, the other is the shifted key function.

For the inverse natural log, press 2nd LN. For the inverse common log, press 2nd LOG.

For example, to calculate the inverse natural log of 2, press 2nd LN 2 ENTER and you'll get about 7.389 .

The trigonometric function [SIN],[COS] and [TAN] have inverse functions arc-sine, arc-cosine, and arc-tangent which are also implemented on scientific calculators. To access the inverse trigonometric functions you use the sequences [SHIFT][SIN], [SHIFT][COS] and [SHIFT][TAN]. The functions are marked on the body of the calculator as [SIN^-1], [COS^-1] and [TAN^-1].
When you use these functions, make sure that the angle unit is correctly set (as required by the problem you are solving) because the angle values returned are in the same unit as the one set.

I depends on what you mean by INV.
On some Chinese (AUREX, Staples) calculators the second function key is marked as INV. If that is the key you have in mind, on Casios it is the [SHIFT] key. On Sharps it is marked as [2ndF]

If you have in mind the multiplicative inverse (the reciprocal) then your calculator has such a key and it is marked [x^-1]. It the key represented on the following table at the intersection of the 2nd column and the 9th line (RECIPROCAL)



If you are thinking about the inverse of functions, then know that each function shares the same physical key as its inverse. For example to access the arcsine (the function inverse of sine) you press the [SHIFT] key before [SIN]. On the calculator you will notice that near the [SIN] key and to its left there is [SIN^-1].

The log base 10 function has an inverse: It is the 10 ^x function. It shares the same physical key with the decimal log. Use [SHIFT][LOG].
The natural log (LN) has an inverse alos and that it the exponentila function e^x. It shares the same physical key with it.

because cotangent, cosecant, and secant are all inverse trig functions, just type in 1 / |shift| trig function and the number if you are finding the angles. It might not be a dedicated button, but its an extra push, and it will give you the same result. *note that I put the shift press only because that is how you get the inverse tan to get an angle from a tan ratio.*