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I have problem in remembering the domain and range values of the inverse trigonometric functions, please someone can explain all these inverse trigonometric functions by the help of unit circle? (or any useful graph)

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    $\begingroup$ the inverse functions, graphically, are symmetric respect to the diagonal $f(x)=x$. Hence you can imagine this symmetric graph for each trigonometric function to see what is the most probable principal branch used to define the inverse. $\endgroup$
    – Masacroso
    Jun 12 '17 at 12:36
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    $\begingroup$ This is a good example of a too broad question. You are basically asking us to explain what ammounts to a couple of pages of mathematics, which is not within the scale of this site. This site is a question-answering site, and for that we need specific questions that can be answered in a reasonable amount of words. For general introductions into large topics like "trigonometric function inverses", textbooks are a better medium for learning. $\endgroup$
    – 5xum
    Jun 12 '17 at 12:46
  • $\begingroup$ Of course, if you read a textbook about the topic and there are still parts which are unknown, that's a perfect time to ask the much more specific question that is bothering you on this site. $\endgroup$
    – 5xum
    Jun 12 '17 at 12:47
  • $\begingroup$ @5xum I see your point and actually agree, but there is upvoted and non-closed precedence for this type of question on this site (here, here, here, here). $\endgroup$
    – Dando18
    Jun 12 '17 at 12:57
  • $\begingroup$ @Dando18 I disagree. The first of your examples is a soft question (this one is very specific, despite being broad). The second is much more specific than this (it gives a specific equation and asks how to remember it). The third is similar to the second, and so is the fourth. None is quite as broad as this question. $\endgroup$
    – 5xum
    Jun 12 '17 at 13:02
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The reason we restrict the domain of trig functions is so that the portion of the function selected becomes $1 - 1$ (passes the horizontal line test) and we can take the inverse. With this in mind, think about the graphs of the original trig functions. Take sine for instance,

                                         we split

Here we restrict the domain to $[\frac{-\pi}{2}, \frac{\pi}{2}]$ to achieve the $1 - 1$ function we need to take the inverse. This sliver has domain $[\frac{-\pi}{2}, \frac{\pi}{2}]$ and range $[-1,1]$ so the inverse, $\arcsin x$, has domain $[-1,1]$ and range $[\frac{-\pi}{2}, \frac{\pi}{2}]$. Repeat this process with each trig function and you'll be able to remember their inverses.

This site here has a good explanation and better graphics that might help you.

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