General Sine and Cosine formula for sum of a finite number of angles

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I was wondering is there is a general formula for $sin(x_1+x_2+x_3+...+x_n)$ as well as for the cosine function. I know that $sin(x_1+x_2)=sin(x_1)cos(x_2)+cos(x_1)sin(x_2)$ and $cos(x_1+x_2)=cos(x_1)cos(x_2)-sin(x_1)sin(x_2)$ But I want to find a general formula for the sum of a finite number of angles for the Sine and the cosine but I didn't noticed any pattern. I suspect that it may have a recursive pattern. Any suggestions and hints (not answers) will be appreciated.




trigonometry




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  • Hint, it is easy to write out using the exponential version of sine and cosine, e to the I Pi beta equals sine theta plus I co
    – ericf
    Jul 28 at 2:41










  • Sorry, my phone went crazy and now I can't edit my previous comment.
    – ericf
    Jul 28 at 2:42










  • @ericf don't worry about it. What are your suggestions?
    – user573497
    Jul 28 at 2:44










  • This is a semi-duplicate of a recent question. (That question only asks about sine.)
    – Blue
    Jul 28 at 2:47










  • Wikipedia has a nice entry on Euler's formula, check it out.
    – ericf
    Jul 28 at 2:47














up vote
1
down vote

favorite
1












I was wondering is there is a general formula for $sin(x_1+x_2+x_3+...+x_n)$ as well as for the cosine function. I know that $sin(x_1+x_2)=sin(x_1)cos(x_2)+cos(x_1)sin(x_2)$ and $cos(x_1+x_2)=cos(x_1)cos(x_2)-sin(x_1)sin(x_2)$ But I want to find a general formula for the sum of a finite number of angles for the Sine and the cosine but I didn't noticed any pattern. I suspect that it may have a recursive pattern. Any suggestions and hints (not answers) will be appreciated.




trigonometry




share|cite|improve this question



















  • Hint, it is easy to write out using the exponential version of sine and cosine, e to the I Pi beta equals sine theta plus I co
    – ericf
    Jul 28 at 2:41










  • Sorry, my phone went crazy and now I can't edit my previous comment.
    – ericf
    Jul 28 at 2:42










  • @ericf don't worry about it. What are your suggestions?
    – user573497
    Jul 28 at 2:44










  • This is a semi-duplicate of a recent question. (That question only asks about sine.)
    – Blue
    Jul 28 at 2:47










  • Wikipedia has a nice entry on Euler's formula, check it out.
    – ericf
    Jul 28 at 2:47












up vote
1
down vote

favorite
1









up vote
1
down vote

favorite
1






1





I was wondering is there is a general formula for $sin(x_1+x_2+x_3+...+x_n)$ as well as for the cosine function. I know that $sin(x_1+x_2)=sin(x_1)cos(x_2)+cos(x_1)sin(x_2)$ and $cos(x_1+x_2)=cos(x_1)cos(x_2)-sin(x_1)sin(x_2)$ But I want to find a general formula for the sum of a finite number of angles for the Sine and the cosine but I didn't noticed any pattern. I suspect that it may have a recursive pattern. Any suggestions and hints (not answers) will be appreciated.




trigonometry




share|cite|improve this question











I was wondering is there is a general formula for $sin(x_1+x_2+x_3+...+x_n)$ as well as for the cosine function. I know that $sin(x_1+x_2)=sin(x_1)cos(x_2)+cos(x_1)sin(x_2)$ and $cos(x_1+x_2)=cos(x_1)cos(x_2)-sin(x_1)sin(x_2)$ But I want to find a general formula for the sum of a finite number of angles for the Sine and the cosine but I didn't noticed any pattern. I suspect that it may have a recursive pattern. Any suggestions and hints (not answers) will be appreciated.





trigonometry





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 28 at 2:37









user573497

2009




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  • Hint, it is easy to write out using the exponential version of sine and cosine, e to the I Pi beta equals sine theta plus I co
    – ericf
    Jul 28 at 2:41










  • Sorry, my phone went crazy and now I can't edit my previous comment.
    – ericf
    Jul 28 at 2:42










  • @ericf don't worry about it. What are your suggestions?
    – user573497
    Jul 28 at 2:44










  • This is a semi-duplicate of a recent question. (That question only asks about sine.)
    – Blue
    Jul 28 at 2:47










  • Wikipedia has a nice entry on Euler's formula, check it out.
    – ericf
    Jul 28 at 2:47
















  • Hint, it is easy to write out using the exponential version of sine and cosine, e to the I Pi beta equals sine theta plus I co
    – ericf
    Jul 28 at 2:41










  • Sorry, my phone went crazy and now I can't edit my previous comment.
    – ericf
    Jul 28 at 2:42










  • @ericf don't worry about it. What are your suggestions?
    – user573497
    Jul 28 at 2:44










  • This is a semi-duplicate of a recent question. (That question only asks about sine.)
    – Blue
    Jul 28 at 2:47










  • Wikipedia has a nice entry on Euler's formula, check it out.
    – ericf
    Jul 28 at 2:47















Hint, it is easy to write out using the exponential version of sine and cosine, e to the I Pi beta equals sine theta plus I co
– ericf
Jul 28 at 2:41




Hint, it is easy to write out using the exponential version of sine and cosine, e to the I Pi beta equals sine theta plus I co
– ericf
Jul 28 at 2:41












Sorry, my phone went crazy and now I can't edit my previous comment.
– ericf
Jul 28 at 2:42




Sorry, my phone went crazy and now I can't edit my previous comment.
– ericf
Jul 28 at 2:42












@ericf don't worry about it. What are your suggestions?
– user573497
Jul 28 at 2:44




@ericf don't worry about it. What are your suggestions?
– user573497
Jul 28 at 2:44












This is a semi-duplicate of a recent question. (That question only asks about sine.)
– Blue
Jul 28 at 2:47




This is a semi-duplicate of a recent question. (That question only asks about sine.)
– Blue
Jul 28 at 2:47












Wikipedia has a nice entry on Euler's formula, check it out.
– ericf
Jul 28 at 2:47




Wikipedia has a nice entry on Euler's formula, check it out.
– ericf
Jul 28 at 2:47















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