Vtyjkyuk

How are arcsine and cosecant different mathematically if cosecant is $frac1sin(x)$ and arcsine is $sin^-1(x)$ which is $frac1sin(x)$? I have tried to find an answer before but nobody explained it well enough.

They are two totally different functions with different domains and ranges and different definitions.

The notation is confusing and it takes a while for students to master the concepts.

Note that the arcsine function or $sin ^-1 x$ as it is common to write is the inverse function under composition not under multiplication.

That is $$sin ( sin ^-1 x )=x$$ is true on the domain of $ sin ^-1 x$

While for $csc x$ the story is different because it is the multiplicative inverse of $sin x$ which is $$ ( csc x)times (sin x) =1$$

$sin^-1(x)$ and $frac1sin x$ are not same

$sin^-1(x)$ is equal to some angle $theta $ such that $sin theta =x $
where $-1le x le 1$ and Range $in [-pi /2,pi /2]$

Here is the graph of $sin^-1(x)$

while $frac1sin x $ is simply 1 divided by $sin x $ $hspace20pt$here $x in R$ and Range $in (-infty,infty)$

Here is the graph of $frac1sin x$

$sin^-1(x)$ doesn't mean $frac1sin(x)$. It means the inverse function of the $sin$ (when restricted to a conventional interval).

If I have some invertible function $f$ on some interval, so that I can write say $y=f(x)$ then we use $f^-1$ to represent the inverse function; we can write $x=f^-1(y)$. However, this can sometimes cause confusion with the multiplicative inverse, which can also sometimes be written using a $cdot^-1$-power.

See https://en.wikipedia.org/wiki/Inverse_function ... (and note the "not to be confused with" link right under the title)

So $arcsin$ is the functional inverse of $sin$ while $operatornamecosec(x)$ is the multiplicative inverse of $sin(x)$.

The functional inverses of trig functions are discussed here:

https://en.wikipedia.org/wiki/Inverse_trigonometric_functions

Mathematicians are often inconsistent; when they write $sin^2(x)$ they mean $(sin(x))^2$ (rather than say functional composition, $sin(sin(x))$), and similarly for other powers - even $-2$ (!) ... but not if that power is $-1$. It's a matter of getting used to the convention, and sticking to $1/sin(x)$ or $(sin(x))^-1$ when you mean to talk about the multiplicative inverse.

To avoid any ambiguity between the reciprocal and the inverse, one may write $$f^-1$$ for the reciprocal and $$mathop f^-1$$ for the inverse.