Learning $arcsin, arccos, arctan$ - how to?












2












$begingroup$


Sorry for asking such question.



I have a very basic understanding of $arcsin, arccos, arctan$ functions. I do know how their graph looks like and not much more beyond that.



Calculate:
Tasks



Which specific keywords should I google to learn how to solve the following tasks? I think those aren't equations (googling 'cyclometric equations' was a dead end). Perhaps you would like to share with some link to a beginner-friendly learning source?



Thank you.










share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    inverse trigonometric functions is what I would call them
    $endgroup$
    – Henry Lee
    Dec 5 '18 at 1:06






  • 1




    $begingroup$
    also a lot of them just require the compound angle formula or some rearrangment to make them a lot easier
    $endgroup$
    – Henry Lee
    Dec 5 '18 at 1:07






  • 1




    $begingroup$
    "cyclometric" is the old name for the inverse trig functions. Any basic book should have sections on how to evaluate these.
    $endgroup$
    – SZN
    Dec 5 '18 at 1:11






  • 1




    $begingroup$
    these are more simple but a start: tutorial.math.lamar.edu/Extras/AlgebraTrigReview/…
    $endgroup$
    – Henry Lee
    Dec 5 '18 at 1:11






  • 1




    $begingroup$
    if you ask specifically about 1 of those questions that you struggle with people will also help
    $endgroup$
    – Henry Lee
    Dec 5 '18 at 1:12
















2












$begingroup$


Sorry for asking such question.



I have a very basic understanding of $arcsin, arccos, arctan$ functions. I do know how their graph looks like and not much more beyond that.



Calculate:
Tasks



Which specific keywords should I google to learn how to solve the following tasks? I think those aren't equations (googling 'cyclometric equations' was a dead end). Perhaps you would like to share with some link to a beginner-friendly learning source?



Thank you.










share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    inverse trigonometric functions is what I would call them
    $endgroup$
    – Henry Lee
    Dec 5 '18 at 1:06






  • 1




    $begingroup$
    also a lot of them just require the compound angle formula or some rearrangment to make them a lot easier
    $endgroup$
    – Henry Lee
    Dec 5 '18 at 1:07






  • 1




    $begingroup$
    "cyclometric" is the old name for the inverse trig functions. Any basic book should have sections on how to evaluate these.
    $endgroup$
    – SZN
    Dec 5 '18 at 1:11






  • 1




    $begingroup$
    these are more simple but a start: tutorial.math.lamar.edu/Extras/AlgebraTrigReview/…
    $endgroup$
    – Henry Lee
    Dec 5 '18 at 1:11






  • 1




    $begingroup$
    if you ask specifically about 1 of those questions that you struggle with people will also help
    $endgroup$
    – Henry Lee
    Dec 5 '18 at 1:12














2












2








2





$begingroup$


Sorry for asking such question.



I have a very basic understanding of $arcsin, arccos, arctan$ functions. I do know how their graph looks like and not much more beyond that.



Calculate:
Tasks



Which specific keywords should I google to learn how to solve the following tasks? I think those aren't equations (googling 'cyclometric equations' was a dead end). Perhaps you would like to share with some link to a beginner-friendly learning source?



Thank you.










share|cite|improve this question









$endgroup$




Sorry for asking such question.



I have a very basic understanding of $arcsin, arccos, arctan$ functions. I do know how their graph looks like and not much more beyond that.



Calculate:
Tasks



Which specific keywords should I google to learn how to solve the following tasks? I think those aren't equations (googling 'cyclometric equations' was a dead end). Perhaps you would like to share with some link to a beginner-friendly learning source?



Thank you.







analysis trigonometry inverse-function






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 5 '18 at 1:05









wenoweno

30511




30511








  • 1




    $begingroup$
    inverse trigonometric functions is what I would call them
    $endgroup$
    – Henry Lee
    Dec 5 '18 at 1:06






  • 1




    $begingroup$
    also a lot of them just require the compound angle formula or some rearrangment to make them a lot easier
    $endgroup$
    – Henry Lee
    Dec 5 '18 at 1:07






  • 1




    $begingroup$
    "cyclometric" is the old name for the inverse trig functions. Any basic book should have sections on how to evaluate these.
    $endgroup$
    – SZN
    Dec 5 '18 at 1:11






  • 1




    $begingroup$
    these are more simple but a start: tutorial.math.lamar.edu/Extras/AlgebraTrigReview/…
    $endgroup$
    – Henry Lee
    Dec 5 '18 at 1:11






  • 1




    $begingroup$
    if you ask specifically about 1 of those questions that you struggle with people will also help
    $endgroup$
    – Henry Lee
    Dec 5 '18 at 1:12














  • 1




    $begingroup$
    inverse trigonometric functions is what I would call them
    $endgroup$
    – Henry Lee
    Dec 5 '18 at 1:06






  • 1




    $begingroup$
    also a lot of them just require the compound angle formula or some rearrangment to make them a lot easier
    $endgroup$
    – Henry Lee
    Dec 5 '18 at 1:07






  • 1




    $begingroup$
    "cyclometric" is the old name for the inverse trig functions. Any basic book should have sections on how to evaluate these.
    $endgroup$
    – SZN
    Dec 5 '18 at 1:11






  • 1




    $begingroup$
    these are more simple but a start: tutorial.math.lamar.edu/Extras/AlgebraTrigReview/…
    $endgroup$
    – Henry Lee
    Dec 5 '18 at 1:11






  • 1




    $begingroup$
    if you ask specifically about 1 of those questions that you struggle with people will also help
    $endgroup$
    – Henry Lee
    Dec 5 '18 at 1:12








1




1




$begingroup$
inverse trigonometric functions is what I would call them
$endgroup$
– Henry Lee
Dec 5 '18 at 1:06




$begingroup$
inverse trigonometric functions is what I would call them
$endgroup$
– Henry Lee
Dec 5 '18 at 1:06




1




1




$begingroup$
also a lot of them just require the compound angle formula or some rearrangment to make them a lot easier
$endgroup$
– Henry Lee
Dec 5 '18 at 1:07




$begingroup$
also a lot of them just require the compound angle formula or some rearrangment to make them a lot easier
$endgroup$
– Henry Lee
Dec 5 '18 at 1:07




1




1




$begingroup$
"cyclometric" is the old name for the inverse trig functions. Any basic book should have sections on how to evaluate these.
$endgroup$
– SZN
Dec 5 '18 at 1:11




$begingroup$
"cyclometric" is the old name for the inverse trig functions. Any basic book should have sections on how to evaluate these.
$endgroup$
– SZN
Dec 5 '18 at 1:11




1




1




$begingroup$
these are more simple but a start: tutorial.math.lamar.edu/Extras/AlgebraTrigReview/…
$endgroup$
– Henry Lee
Dec 5 '18 at 1:11




$begingroup$
these are more simple but a start: tutorial.math.lamar.edu/Extras/AlgebraTrigReview/…
$endgroup$
– Henry Lee
Dec 5 '18 at 1:11




1




1




$begingroup$
if you ask specifically about 1 of those questions that you struggle with people will also help
$endgroup$
– Henry Lee
Dec 5 '18 at 1:12




$begingroup$
if you ask specifically about 1 of those questions that you struggle with people will also help
$endgroup$
– Henry Lee
Dec 5 '18 at 1:12










3 Answers
3






active

oldest

votes


















2












$begingroup$

I would say there are three things you are expected to do on this list. One is to know the trig functions of special angles, so for 4 you should know that $tan frac pi 4=1,$ so $arctan 1=frac pi 4$ Watch out for the ranges specified for the inverse trig functions. Second is that $sin(arcsin (x))=x$. When you have $arcsin (sin(x))$ you may be shifted by factors of $pi$. Finally when you have $sin(arccos(frac 13))$ draw a right triangle with $cos$ of one angle $frac 13$, so it is a $1-sqrt 8-3$ triangle and find the sine of the angle, here $frac 13sqrt 8$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Hey, thanks. I understood everything except for: "When you have arcsin(sin(x)) you may be shifted by factors of π". Would you give a little bit more insight on this?
    $endgroup$
    – weno
    Dec 5 '18 at 1:19












  • $begingroup$
    The range of $arcsin$ is defined as $[-frac pi 2,frac pi 2]$, so if we are asked for $arcsin(sin(frac {9pi}4))$ the answer is $frac pi 4,$ not $frac {9 pi}4$
    $endgroup$
    – Ross Millikan
    Dec 5 '18 at 1:22












  • $begingroup$
    Those indeed are the tricky questions which I was referring to!
    $endgroup$
    – gimusi
    Dec 5 '18 at 1:23










  • $begingroup$
    I don't know which post should I accept, since you both helped me a lot. I'll do a random.org roll.
    $endgroup$
    – weno
    Dec 5 '18 at 1:29



















2












$begingroup$

Some good reference as summary from Wikipedia are




  • List of trigonometric identities


and also




  • Inverse trigonometric functions


Some exercises given requires only to calculate the value for the functions at some point, other else are more tricky and you need to acquire a deeply understanding of the matter.



Refer also to the related




  • Reference Request: Book for Trigonometry and Geometry


  • Good book on advanced trig







share|cite|improve this answer











$endgroup$













  • $begingroup$
    That should help, thanks.
    $endgroup$
    – weno
    Dec 5 '18 at 1:14










  • $begingroup$
    Do not hesitate to ask for any exercise in particular! Bye
    $endgroup$
    – gimusi
    Dec 5 '18 at 1:15










  • $begingroup$
    I would, but unless I made any significant progress myself, I don't think anyone will solve that for me. :)
    $endgroup$
    – weno
    Dec 5 '18 at 1:17










  • $begingroup$
    That’s nice but in case of doubt I would happy to check that with you. Bye
    $endgroup$
    – gimusi
    Dec 5 '18 at 1:19



















1












$begingroup$

This lecture has been particularly helpful on understanding the subject:



https://academics.utep.edu/Portals/1788/CALCULUS%20MATERIAL/4_7%20INVERSE%20TRIG%20FNS.pdf






share|cite|improve this answer









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    3 Answers
    3






    active

    oldest

    votes








    3 Answers
    3






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    2












    $begingroup$

    I would say there are three things you are expected to do on this list. One is to know the trig functions of special angles, so for 4 you should know that $tan frac pi 4=1,$ so $arctan 1=frac pi 4$ Watch out for the ranges specified for the inverse trig functions. Second is that $sin(arcsin (x))=x$. When you have $arcsin (sin(x))$ you may be shifted by factors of $pi$. Finally when you have $sin(arccos(frac 13))$ draw a right triangle with $cos$ of one angle $frac 13$, so it is a $1-sqrt 8-3$ triangle and find the sine of the angle, here $frac 13sqrt 8$.






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      Hey, thanks. I understood everything except for: "When you have arcsin(sin(x)) you may be shifted by factors of π". Would you give a little bit more insight on this?
      $endgroup$
      – weno
      Dec 5 '18 at 1:19












    • $begingroup$
      The range of $arcsin$ is defined as $[-frac pi 2,frac pi 2]$, so if we are asked for $arcsin(sin(frac {9pi}4))$ the answer is $frac pi 4,$ not $frac {9 pi}4$
      $endgroup$
      – Ross Millikan
      Dec 5 '18 at 1:22












    • $begingroup$
      Those indeed are the tricky questions which I was referring to!
      $endgroup$
      – gimusi
      Dec 5 '18 at 1:23










    • $begingroup$
      I don't know which post should I accept, since you both helped me a lot. I'll do a random.org roll.
      $endgroup$
      – weno
      Dec 5 '18 at 1:29
















    2












    $begingroup$

    I would say there are three things you are expected to do on this list. One is to know the trig functions of special angles, so for 4 you should know that $tan frac pi 4=1,$ so $arctan 1=frac pi 4$ Watch out for the ranges specified for the inverse trig functions. Second is that $sin(arcsin (x))=x$. When you have $arcsin (sin(x))$ you may be shifted by factors of $pi$. Finally when you have $sin(arccos(frac 13))$ draw a right triangle with $cos$ of one angle $frac 13$, so it is a $1-sqrt 8-3$ triangle and find the sine of the angle, here $frac 13sqrt 8$.






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      Hey, thanks. I understood everything except for: "When you have arcsin(sin(x)) you may be shifted by factors of π". Would you give a little bit more insight on this?
      $endgroup$
      – weno
      Dec 5 '18 at 1:19












    • $begingroup$
      The range of $arcsin$ is defined as $[-frac pi 2,frac pi 2]$, so if we are asked for $arcsin(sin(frac {9pi}4))$ the answer is $frac pi 4,$ not $frac {9 pi}4$
      $endgroup$
      – Ross Millikan
      Dec 5 '18 at 1:22












    • $begingroup$
      Those indeed are the tricky questions which I was referring to!
      $endgroup$
      – gimusi
      Dec 5 '18 at 1:23










    • $begingroup$
      I don't know which post should I accept, since you both helped me a lot. I'll do a random.org roll.
      $endgroup$
      – weno
      Dec 5 '18 at 1:29














    2












    2








    2





    $begingroup$

    I would say there are three things you are expected to do on this list. One is to know the trig functions of special angles, so for 4 you should know that $tan frac pi 4=1,$ so $arctan 1=frac pi 4$ Watch out for the ranges specified for the inverse trig functions. Second is that $sin(arcsin (x))=x$. When you have $arcsin (sin(x))$ you may be shifted by factors of $pi$. Finally when you have $sin(arccos(frac 13))$ draw a right triangle with $cos$ of one angle $frac 13$, so it is a $1-sqrt 8-3$ triangle and find the sine of the angle, here $frac 13sqrt 8$.






    share|cite|improve this answer











    $endgroup$



    I would say there are three things you are expected to do on this list. One is to know the trig functions of special angles, so for 4 you should know that $tan frac pi 4=1,$ so $arctan 1=frac pi 4$ Watch out for the ranges specified for the inverse trig functions. Second is that $sin(arcsin (x))=x$. When you have $arcsin (sin(x))$ you may be shifted by factors of $pi$. Finally when you have $sin(arccos(frac 13))$ draw a right triangle with $cos$ of one angle $frac 13$, so it is a $1-sqrt 8-3$ triangle and find the sine of the angle, here $frac 13sqrt 8$.







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited Dec 5 '18 at 1:56

























    answered Dec 5 '18 at 1:17









    Ross MillikanRoss Millikan

    295k23198371




    295k23198371












    • $begingroup$
      Hey, thanks. I understood everything except for: "When you have arcsin(sin(x)) you may be shifted by factors of π". Would you give a little bit more insight on this?
      $endgroup$
      – weno
      Dec 5 '18 at 1:19












    • $begingroup$
      The range of $arcsin$ is defined as $[-frac pi 2,frac pi 2]$, so if we are asked for $arcsin(sin(frac {9pi}4))$ the answer is $frac pi 4,$ not $frac {9 pi}4$
      $endgroup$
      – Ross Millikan
      Dec 5 '18 at 1:22












    • $begingroup$
      Those indeed are the tricky questions which I was referring to!
      $endgroup$
      – gimusi
      Dec 5 '18 at 1:23










    • $begingroup$
      I don't know which post should I accept, since you both helped me a lot. I'll do a random.org roll.
      $endgroup$
      – weno
      Dec 5 '18 at 1:29


















    • $begingroup$
      Hey, thanks. I understood everything except for: "When you have arcsin(sin(x)) you may be shifted by factors of π". Would you give a little bit more insight on this?
      $endgroup$
      – weno
      Dec 5 '18 at 1:19












    • $begingroup$
      The range of $arcsin$ is defined as $[-frac pi 2,frac pi 2]$, so if we are asked for $arcsin(sin(frac {9pi}4))$ the answer is $frac pi 4,$ not $frac {9 pi}4$
      $endgroup$
      – Ross Millikan
      Dec 5 '18 at 1:22












    • $begingroup$
      Those indeed are the tricky questions which I was referring to!
      $endgroup$
      – gimusi
      Dec 5 '18 at 1:23










    • $begingroup$
      I don't know which post should I accept, since you both helped me a lot. I'll do a random.org roll.
      $endgroup$
      – weno
      Dec 5 '18 at 1:29
















    $begingroup$
    Hey, thanks. I understood everything except for: "When you have arcsin(sin(x)) you may be shifted by factors of π". Would you give a little bit more insight on this?
    $endgroup$
    – weno
    Dec 5 '18 at 1:19






    $begingroup$
    Hey, thanks. I understood everything except for: "When you have arcsin(sin(x)) you may be shifted by factors of π". Would you give a little bit more insight on this?
    $endgroup$
    – weno
    Dec 5 '18 at 1:19














    $begingroup$
    The range of $arcsin$ is defined as $[-frac pi 2,frac pi 2]$, so if we are asked for $arcsin(sin(frac {9pi}4))$ the answer is $frac pi 4,$ not $frac {9 pi}4$
    $endgroup$
    – Ross Millikan
    Dec 5 '18 at 1:22






    $begingroup$
    The range of $arcsin$ is defined as $[-frac pi 2,frac pi 2]$, so if we are asked for $arcsin(sin(frac {9pi}4))$ the answer is $frac pi 4,$ not $frac {9 pi}4$
    $endgroup$
    – Ross Millikan
    Dec 5 '18 at 1:22














    $begingroup$
    Those indeed are the tricky questions which I was referring to!
    $endgroup$
    – gimusi
    Dec 5 '18 at 1:23




    $begingroup$
    Those indeed are the tricky questions which I was referring to!
    $endgroup$
    – gimusi
    Dec 5 '18 at 1:23












    $begingroup$
    I don't know which post should I accept, since you both helped me a lot. I'll do a random.org roll.
    $endgroup$
    – weno
    Dec 5 '18 at 1:29




    $begingroup$
    I don't know which post should I accept, since you both helped me a lot. I'll do a random.org roll.
    $endgroup$
    – weno
    Dec 5 '18 at 1:29











    2












    $begingroup$

    Some good reference as summary from Wikipedia are




    • List of trigonometric identities


    and also




    • Inverse trigonometric functions


    Some exercises given requires only to calculate the value for the functions at some point, other else are more tricky and you need to acquire a deeply understanding of the matter.



    Refer also to the related




    • Reference Request: Book for Trigonometry and Geometry


    • Good book on advanced trig







    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      That should help, thanks.
      $endgroup$
      – weno
      Dec 5 '18 at 1:14










    • $begingroup$
      Do not hesitate to ask for any exercise in particular! Bye
      $endgroup$
      – gimusi
      Dec 5 '18 at 1:15










    • $begingroup$
      I would, but unless I made any significant progress myself, I don't think anyone will solve that for me. :)
      $endgroup$
      – weno
      Dec 5 '18 at 1:17










    • $begingroup$
      That’s nice but in case of doubt I would happy to check that with you. Bye
      $endgroup$
      – gimusi
      Dec 5 '18 at 1:19
















    2












    $begingroup$

    Some good reference as summary from Wikipedia are




    • List of trigonometric identities


    and also




    • Inverse trigonometric functions


    Some exercises given requires only to calculate the value for the functions at some point, other else are more tricky and you need to acquire a deeply understanding of the matter.



    Refer also to the related




    • Reference Request: Book for Trigonometry and Geometry


    • Good book on advanced trig







    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      That should help, thanks.
      $endgroup$
      – weno
      Dec 5 '18 at 1:14










    • $begingroup$
      Do not hesitate to ask for any exercise in particular! Bye
      $endgroup$
      – gimusi
      Dec 5 '18 at 1:15










    • $begingroup$
      I would, but unless I made any significant progress myself, I don't think anyone will solve that for me. :)
      $endgroup$
      – weno
      Dec 5 '18 at 1:17










    • $begingroup$
      That’s nice but in case of doubt I would happy to check that with you. Bye
      $endgroup$
      – gimusi
      Dec 5 '18 at 1:19














    2












    2








    2





    $begingroup$

    Some good reference as summary from Wikipedia are




    • List of trigonometric identities


    and also




    • Inverse trigonometric functions


    Some exercises given requires only to calculate the value for the functions at some point, other else are more tricky and you need to acquire a deeply understanding of the matter.



    Refer also to the related




    • Reference Request: Book for Trigonometry and Geometry


    • Good book on advanced trig







    share|cite|improve this answer











    $endgroup$



    Some good reference as summary from Wikipedia are




    • List of trigonometric identities


    and also




    • Inverse trigonometric functions


    Some exercises given requires only to calculate the value for the functions at some point, other else are more tricky and you need to acquire a deeply understanding of the matter.



    Refer also to the related




    • Reference Request: Book for Trigonometry and Geometry


    • Good book on advanced trig








    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited Dec 5 '18 at 1:14

























    answered Dec 5 '18 at 1:10









    gimusigimusi

    92.8k84494




    92.8k84494












    • $begingroup$
      That should help, thanks.
      $endgroup$
      – weno
      Dec 5 '18 at 1:14










    • $begingroup$
      Do not hesitate to ask for any exercise in particular! Bye
      $endgroup$
      – gimusi
      Dec 5 '18 at 1:15










    • $begingroup$
      I would, but unless I made any significant progress myself, I don't think anyone will solve that for me. :)
      $endgroup$
      – weno
      Dec 5 '18 at 1:17










    • $begingroup$
      That’s nice but in case of doubt I would happy to check that with you. Bye
      $endgroup$
      – gimusi
      Dec 5 '18 at 1:19


















    • $begingroup$
      That should help, thanks.
      $endgroup$
      – weno
      Dec 5 '18 at 1:14










    • $begingroup$
      Do not hesitate to ask for any exercise in particular! Bye
      $endgroup$
      – gimusi
      Dec 5 '18 at 1:15










    • $begingroup$
      I would, but unless I made any significant progress myself, I don't think anyone will solve that for me. :)
      $endgroup$
      – weno
      Dec 5 '18 at 1:17










    • $begingroup$
      That’s nice but in case of doubt I would happy to check that with you. Bye
      $endgroup$
      – gimusi
      Dec 5 '18 at 1:19
















    $begingroup$
    That should help, thanks.
    $endgroup$
    – weno
    Dec 5 '18 at 1:14




    $begingroup$
    That should help, thanks.
    $endgroup$
    – weno
    Dec 5 '18 at 1:14












    $begingroup$
    Do not hesitate to ask for any exercise in particular! Bye
    $endgroup$
    – gimusi
    Dec 5 '18 at 1:15




    $begingroup$
    Do not hesitate to ask for any exercise in particular! Bye
    $endgroup$
    – gimusi
    Dec 5 '18 at 1:15












    $begingroup$
    I would, but unless I made any significant progress myself, I don't think anyone will solve that for me. :)
    $endgroup$
    – weno
    Dec 5 '18 at 1:17




    $begingroup$
    I would, but unless I made any significant progress myself, I don't think anyone will solve that for me. :)
    $endgroup$
    – weno
    Dec 5 '18 at 1:17












    $begingroup$
    That’s nice but in case of doubt I would happy to check that with you. Bye
    $endgroup$
    – gimusi
    Dec 5 '18 at 1:19




    $begingroup$
    That’s nice but in case of doubt I would happy to check that with you. Bye
    $endgroup$
    – gimusi
    Dec 5 '18 at 1:19











    1












    $begingroup$

    This lecture has been particularly helpful on understanding the subject:



    https://academics.utep.edu/Portals/1788/CALCULUS%20MATERIAL/4_7%20INVERSE%20TRIG%20FNS.pdf






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      This lecture has been particularly helpful on understanding the subject:



      https://academics.utep.edu/Portals/1788/CALCULUS%20MATERIAL/4_7%20INVERSE%20TRIG%20FNS.pdf






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        This lecture has been particularly helpful on understanding the subject:



        https://academics.utep.edu/Portals/1788/CALCULUS%20MATERIAL/4_7%20INVERSE%20TRIG%20FNS.pdf






        share|cite|improve this answer









        $endgroup$



        This lecture has been particularly helpful on understanding the subject:



        https://academics.utep.edu/Portals/1788/CALCULUS%20MATERIAL/4_7%20INVERSE%20TRIG%20FNS.pdf







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 5 '18 at 3:35









        wenoweno

        30511




        30511






























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