Is it possible to express values such as $\sin^{-1}(\pi/12)$ without inverse trig functions?

Question

Pretty self explanatory.

If I had for example something like $\sin^{-1}(\pi/12)$ in an expression, is it ever possible to express that expression without inverse trig functions?

Answer 1

If you have fixed arguments, the inverse trigonometric function (or any other, for that matter) is just a fixed value. Finding out what the value is (giving it a "name" that isn't just the expression to be replaced) might or might not be possible or simple to do. Or even desirable, if e.g. you want to apply trigonometric functions to your expression the inverse trigonometric functions can point to ways to simplify the result.

Answer 2

You can give your number a name or represent it explicitly e.g. as value of another function or of an integral, as limit of a series, or implicitly as solution of an equation.
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The numbers that are explicitly representable by finite terms are the explicit ones among the closed-form numbers.

Because $\sin^{-1}(\frac{\pi}{12})$ is the value of a named function ($\sin^{-1}$), it is an explicit closed-form number.
Because $\sin^{-1}$ is an explicit elementary function, $\sin^{-1}(\frac{\pi}{12})$ is an explicit elementary number, an EL-number (see below: [Chow 1999]).

Let $z_0\in\mathbb{C}$ and let $c$ denote a closed-form number. Your problem has the form:

$$\sin^{-1}(z_0)=c,$$

or

$$\sin(c)=z_0.$$

Your $z_0$ is also an explicit closed-form number, in your example in particular also an explicit elementary number.

1.) Algebraic numbers

a) If your $z_0$ is an algebraic number, look for the algebraic points of your function ($\sin^{-1}$) (or for the algebraic points of the inverse of your function ($\sin$)).

b) Otherwise look for the algebraic values of your function ($\sin^{-1}$).

We know, the explicitly representable algebraic numbers are the numbers that can be represented by radical expressions.
But $\sin^{-1}(\frac{\pi}{12})$ is not an explicit algebraic number.

2.) Elementary numbers

$\sin^{-1}(z)$ is an elementary function.

The function term of an elementary function can be represented as a term that contains only the variable and $\exp$, $\log$ and/or algebraic terms.

$$\sin^{-1}(z)=-i\log(\sqrt{1-z^2}+iz)$$

$$\sin^{-1}(\frac{\pi}{12})=-i\log(\sqrt{1-\left(\frac{\pi}{12}\right)^2}+i\frac{\pi}{12})$$

[Chow 1999] Chow, T.: What is a closed-form number. Am. Math. Monthly 106 (1999) (5) 440-448