Separate $sin^-1 (cos theta + i sin theta)$ into real and imaginary parts [on hold]

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I tried it but it became horribly long. Please suggest a concise method to solve it.






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put on hold as off-topic by Did, Xander Henderson, hardmath, Micah, Key Flex 11 hours ago


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  • If you "tried it but it became horribly long", some details of how you approached this would make a welcome improvement. In its current form the body of the Question does not contain a problem statement, and it is not a good practice to rely solely on the title for the problem setup and goal. Please edit.
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I tried it but it became horribly long. Please suggest a concise method to solve it.






trigonometry







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put on hold as off-topic by Did, Xander Henderson, hardmath, Micah, Key Flex 11 hours ago


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Xander Henderson, hardmath, Micah, Key Flex
If this question can be reworded to fit the rules in the help center, please edit the question.












  • If you "tried it but it became horribly long", some details of how you approached this would make a welcome improvement. In its current form the body of the Question does not contain a problem statement, and it is not a good practice to rely solely on the title for the problem setup and goal. Please edit.
    – hardmath
    14 hours ago












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I tried it but it became horribly long. Please suggest a concise method to solve it.






trigonometry







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I tried it but it became horribly long. Please suggest a concise method to solve it.






trigonometry




trigonometry






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edited Sep 9 at 14:01









amWhy

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asked Sep 9 at 13:44









Pratim Das

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put on hold as off-topic by Did, Xander Henderson, hardmath, Micah, Key Flex 11 hours ago


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Xander Henderson, hardmath, Micah, Key Flex
If this question can be reworded to fit the rules in the help center, please edit the question.




put on hold as off-topic by Did, Xander Henderson, hardmath, Micah, Key Flex 11 hours ago


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Xander Henderson, hardmath, Micah, Key Flex
If this question can be reworded to fit the rules in the help center, please edit the question.











  • If you "tried it but it became horribly long", some details of how you approached this would make a welcome improvement. In its current form the body of the Question does not contain a problem statement, and it is not a good practice to rely solely on the title for the problem setup and goal. Please edit.
    – hardmath
    14 hours ago
















  • If you "tried it but it became horribly long", some details of how you approached this would make a welcome improvement. In its current form the body of the Question does not contain a problem statement, and it is not a good practice to rely solely on the title for the problem setup and goal. Please edit.
    – hardmath
    14 hours ago















If you "tried it but it became horribly long", some details of how you approached this would make a welcome improvement. In its current form the body of the Question does not contain a problem statement, and it is not a good practice to rely solely on the title for the problem setup and goal. Please edit.
– hardmath
14 hours ago




If you "tried it but it became horribly long", some details of how you approached this would make a welcome improvement. In its current form the body of the Question does not contain a problem statement, and it is not a good practice to rely solely on the title for the problem setup and goal. Please edit.
– hardmath
14 hours ago










1 Answer
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Let $sin^-1(costheta+isintheta)$ be $a+bi$. Then



$costheta+isintheta=sin(a+bi)=sin acos (bi) + cos asin(bi)$



Now just express $cos(bi)$ and $sin(bi)$ in another way to get the real and imaginary parts.






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    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    2
    down vote













    Let $sin^-1(costheta+isintheta)$ be $a+bi$. Then



    $costheta+isintheta=sin(a+bi)=sin acos (bi) + cos asin(bi)$



    Now just express $cos(bi)$ and $sin(bi)$ in another way to get the real and imaginary parts.






    share|cite|improve this answer
























      up vote
      2
      down vote













      Let $sin^-1(costheta+isintheta)$ be $a+bi$. Then



      $costheta+isintheta=sin(a+bi)=sin acos (bi) + cos asin(bi)$



      Now just express $cos(bi)$ and $sin(bi)$ in another way to get the real and imaginary parts.






      share|cite|improve this answer






















        up vote
        2
        down vote










        up vote
        2
        down vote









        Let $sin^-1(costheta+isintheta)$ be $a+bi$. Then



        $costheta+isintheta=sin(a+bi)=sin acos (bi) + cos asin(bi)$



        Now just express $cos(bi)$ and $sin(bi)$ in another way to get the real and imaginary parts.






        share|cite|improve this answer












        Let $sin^-1(costheta+isintheta)$ be $a+bi$. Then



        $costheta+isintheta=sin(a+bi)=sin acos (bi) + cos asin(bi)$



        Now just express $cos(bi)$ and $sin(bi)$ in another way to get the real and imaginary parts.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Sep 9 at 13:53









        TonyK

        38.9k348127




        38.9k348127












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