How do we evaluate inverse trigonometric relations and functions?

How do we evaluate inverse trigonometric relations and functions? 

Okay, so normal trigonometric functions are like sin(x), cos(x), and tan(x). And some added ones are sec(x), csc(x), and cot(x).

But now there's inverse functions that are just inverse of these regular functions. There's sin⁻¹(x), cos⁻¹(x), and tan⁻¹(x). As for the co-functions there are sec⁻¹(x), csc⁻¹(x), and cot⁻¹(x).

You will used these inverses when looking at arc lines, which are reflected from their original line over line y=x. For example for sin(x)

The red line is sin(x) and the blue line the the inverse sin⁻¹(x).
The green line shows y=x and how both lines reflect
over each other but cross each other's paths.

For arc lines you write them a little differently. For sin⁻¹(x) it's be written as arcsin(x). Here's a chart I made so you'd remember. 

So let's try a practice problem.

Let's find the arcsin (1/2) + arctan(1)

We know that arcsin is sin⁻¹ and arctan is tan⁻¹ , so we can just rewrite this.

sin⁻¹ (1/2) + tan⁻¹ (1)

Plugging that into your calculator you should get sin⁻¹ (1/2) = 30 and tan⁻¹ (1) = 45

So simplifying the problem :

30 + 45 = 75°

Sources:

1. John Schnatterly: johnschnatterly.blogspot.com