Intuition for trigonometric function of complex number [on hold]

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How can I intuitively understand trigonometric functions on a complex angle like what does $cos(theta)$ even mean when $theta$ is complex. How can I represent this as I would have done showing a circle of radius $R$ whose any point can be represented as $P (R cos(theta), R sin(theta)) $ if $theta$ would have been real.




trigonometry complex-numbers




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put on hold as off-topic by amWhy, Taroccoesbrocco, Jyrki Lahtonen, John Ma, user 108128 2 days 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." – amWhy, Taroccoesbrocco, Jyrki Lahtonen, John Ma, user 108128
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    In a case like this, I would recommend trying not for intuition, but for experience. That is, work out some examples. Make sure you can see that even for complex $theta$, you get the relation $costheta=cos(theta+2pi)$.
    – Lubin
    Aug 3 at 17:56















up vote
0
down vote

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How can I intuitively understand trigonometric functions on a complex angle like what does $cos(theta)$ even mean when $theta$ is complex. How can I represent this as I would have done showing a circle of radius $R$ whose any point can be represented as $P (R cos(theta), R sin(theta)) $ if $theta$ would have been real.




trigonometry complex-numbers




share|cite|improve this question













put on hold as off-topic by amWhy, Taroccoesbrocco, Jyrki Lahtonen, John Ma, user 108128 2 days 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." – amWhy, Taroccoesbrocco, Jyrki Lahtonen, John Ma, user 108128
If this question can be reworded to fit the rules in the help center, please edit the question.








  • 2




    In a case like this, I would recommend trying not for intuition, but for experience. That is, work out some examples. Make sure you can see that even for complex $theta$, you get the relation $costheta=cos(theta+2pi)$.
    – Lubin
    Aug 3 at 17:56













up vote
0
down vote

favorite









up vote
0
down vote

favorite











How can I intuitively understand trigonometric functions on a complex angle like what does $cos(theta)$ even mean when $theta$ is complex. How can I represent this as I would have done showing a circle of radius $R$ whose any point can be represented as $P (R cos(theta), R sin(theta)) $ if $theta$ would have been real.




trigonometry complex-numbers




share|cite|improve this question













How can I intuitively understand trigonometric functions on a complex angle like what does $cos(theta)$ even mean when $theta$ is complex. How can I represent this as I would have done showing a circle of radius $R$ whose any point can be represented as $P (R cos(theta), R sin(theta)) $ if $theta$ would have been real.





trigonometry complex-numbers





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Aug 3 at 17:18









Adrian Keister

3,49321433




3,49321433









asked Aug 3 at 17:11









Vishal Goyal

42




42




put on hold as off-topic by amWhy, Taroccoesbrocco, Jyrki Lahtonen, John Ma, user 108128 2 days 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." – amWhy, Taroccoesbrocco, Jyrki Lahtonen, John Ma, user 108128
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 amWhy, Taroccoesbrocco, Jyrki Lahtonen, John Ma, user 108128 2 days 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." – amWhy, Taroccoesbrocco, Jyrki Lahtonen, John Ma, user 108128
If this question can be reworded to fit the rules in the help center, please edit the question.







  • 2




    In a case like this, I would recommend trying not for intuition, but for experience. That is, work out some examples. Make sure you can see that even for complex $theta$, you get the relation $costheta=cos(theta+2pi)$.
    – Lubin
    Aug 3 at 17:56













  • 2




    In a case like this, I would recommend trying not for intuition, but for experience. That is, work out some examples. Make sure you can see that even for complex $theta$, you get the relation $costheta=cos(theta+2pi)$.
    – Lubin
    Aug 3 at 17:56








2




2




In a case like this, I would recommend trying not for intuition, but for experience. That is, work out some examples. Make sure you can see that even for complex $theta$, you get the relation $costheta=cos(theta+2pi)$.
– Lubin
Aug 3 at 17:56





In a case like this, I would recommend trying not for intuition, but for experience. That is, work out some examples. Make sure you can see that even for complex $theta$, you get the relation $costheta=cos(theta+2pi)$.
– Lubin
Aug 3 at 17:56











1 Answer
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We can refer to the expression obtained by Euler's formula



$$cos theta=frace^itheta+e^-itheta2$$



but of course we can't visualize that function on the complex plane in a easy way such as for the circle in the real case.



What we can do is to plot a surface for $operatornameRe(cos theta)$, $operatornameIm(cos theta)$ or $|cos theta|$ to obtain the following plots.






share|cite|improve this answer






























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    0
    down vote













    We can refer to the expression obtained by Euler's formula



    $$cos theta=frace^itheta+e^-itheta2$$



    but of course we can't visualize that function on the complex plane in a easy way such as for the circle in the real case.



    What we can do is to plot a surface for $operatornameRe(cos theta)$, $operatornameIm(cos theta)$ or $|cos theta|$ to obtain the following plots.






    share|cite|improve this answer



























      up vote
      0
      down vote













      We can refer to the expression obtained by Euler's formula



      $$cos theta=frace^itheta+e^-itheta2$$



      but of course we can't visualize that function on the complex plane in a easy way such as for the circle in the real case.



      What we can do is to plot a surface for $operatornameRe(cos theta)$, $operatornameIm(cos theta)$ or $|cos theta|$ to obtain the following plots.






      share|cite|improve this answer

























        up vote
        0
        down vote










        up vote
        0
        down vote









        We can refer to the expression obtained by Euler's formula



        $$cos theta=frace^itheta+e^-itheta2$$



        but of course we can't visualize that function on the complex plane in a easy way such as for the circle in the real case.



        What we can do is to plot a surface for $operatornameRe(cos theta)$, $operatornameIm(cos theta)$ or $|cos theta|$ to obtain the following plots.






        share|cite|improve this answer















        We can refer to the expression obtained by Euler's formula



        $$cos theta=frace^itheta+e^-itheta2$$



        but of course we can't visualize that function on the complex plane in a easy way such as for the circle in the real case.



        What we can do is to plot a surface for $operatornameRe(cos theta)$, $operatornameIm(cos theta)$ or $|cos theta|$ to obtain the following plots.







        share|cite|improve this answer















        share|cite|improve this answer



        share|cite|improve this answer








        edited Aug 3 at 17:22


























        answered Aug 3 at 17:20









        gimusi

        63.7k73480




        63.7k73480












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