Hello,
\[ 2^{-1} = \frac{1}{2} = 0.5 \] Or so I was taught in algebra. \[ sin(x) \] where x = 30 is 0.5. So I expected \[ sin^{-1}(x) \] where X is 30 to equal 2. But that's not the case. My TI-89 says that the result is undefined.
However if I do \[ \frac{1}{sin(x)} \] I get 2 as expected. And the result the book I'm reading says I should get for cosec.
I can do similarly for \[ cos^{-1}(x) \] and \[ tan^{-1}(x) \] using various values for x.

So why is a trig function to the negative one not divided by one?
Are there cosec, cotangent, and secant buttons on calculators?

Thanks!

 

The meaning of the notation is not the reciprocal of the function, but its inverse. This is a common mathematical notation. The prefix "arc" is another way of denoting the inverse. Thus

\( sin^{-1}(x)\;=\;arcsin(x) \)

The meaning of the inverse sine is to take an argument in [-1,...1] and return an angle in radians, degrees or grads.

The proper notation for reciprocal would be:

\( (sin(x))^{-1}\;=\;\cfrac{1}{sin(x)} \)