On which quadrants are each of the six inverse trig functions defined?

[smashbrohamme]

I have researched this area a little bit and now I am a little worried because three different websites have gave me three different answers. Some functions matched, but others didn't.

My general consensus is
inverse Sin= 1 and 4 quad
inverse Cos= 1 and 2 quad
inverse tan= 1 and 4 quad
inverse cotangent = 1 and 2 quad
inverse secant= 1 and 2 quad
inverse cosecant= 1 and 4 quad

More importantly is there a quick way to verify this in your calculator?

I tried giving random angles for the inverse functions in my calculator to the so called undefined quadrants for the directed inverse functions but I am not getting a light bulb here.

So maybe a little help?

 

[Infinitum]

Your quadrants for the respective functions are correct, but just remember that arcsec isn't defined at $\pi/2$, and similarly for arccosec.

These are the principal values of the respective inverse functions. Meaning, for any x, arcsin(x) will definitely give you an answer in $(-\pi/2, \pi/2)$ (it is the range for the function). It could also give an answer other other than this range, but then the function itself wouldn't be defined as you need to have a unique element in the range satisfying x. The quadrants you wrote are chosen to be the principle ranges, to define the inverse trigonometric functions.

 

[smashbrohamme]

can you give me an example on how to prove this?

I am having a hard time grasping this.

 

[Infinitum]

can you give me an example on how to prove this?

I am having a hard time grasping this.

Prove what?

As I said, those above principle values were chosen to be the ranges, as they give values for every x. This was done to define the inverse function as functions cannot be have multiple values for the same x.

Read through this to get a better idea.
http://oakroadsystems.com/twt/inverse.htm