Find sum from k equals 1 to 6 of open parentheses sin fraction numerator 2 pi k over denominator 7 end fraction space minus space i space cos space fraction numerator 2 pi k over denominator 7 end fraction close parentheses
where i is complex number.
 

Asked by yashbbelkhedkar1 | 2nd Oct, 2014, 07:05: PM

Expert Answer:

e to the power of i theta end exponent equals cos space theta plus i space sin space theta A l s o comma space cos space theta equals fraction numerator e to the power of i theta end exponent plus e to the power of minus i theta end exponent over denominator 2 end fraction rightwards double arrow i cos space theta equals fraction numerator i left parenthesis e to the power of i theta end exponent plus e to the power of minus i theta end exponent right parenthesis over denominator 2 end fraction space left square bracket M u l t i p l y i n g space b o t h space s i d e s space b y space i right square bracket rightwards double arrow i cos space theta equals fraction numerator i squared left parenthesis e to the power of i theta end exponent plus e to the power of minus i theta end exponent right parenthesis over denominator 2 i end fraction left square bracket M u t i l p y i n g space b o t h space n u m e r a t o r space a n d space d e n o m i n a t o r space b y space i space o n space r i g h t space h a n d space s i d e right square bracket rightwards double arrow i cos space theta equals minus fraction numerator left parenthesis e to the power of i theta end exponent plus e to the power of minus i theta end exponent right parenthesis over denominator 2 i end fraction...... left parenthesis i right parenthesis sin space theta equals fraction numerator e to the power of i theta end exponent minus e to the power of i theta end exponent over denominator 2 i end fraction....... left parenthesis i i right parenthesis left parenthesis i right parenthesis minus left parenthesis i i right parenthesis comma sin space theta minus i cos space theta equals fraction numerator e to the power of i theta end exponent minus e to the power of i theta end exponent over denominator 2 i end fraction plus fraction numerator left parenthesis e to the power of i theta end exponent plus e to the power of minus i theta end exponent right parenthesis over denominator 2 i end fraction rightwards double arrow sin space theta minus i cos space theta equals fraction numerator e to the power of i theta end exponent over denominator 2 i end fraction plus fraction numerator e to the power of i theta end exponent over denominator 2 i end fraction rightwards double arrow sin space theta minus i cos space theta equals fraction numerator 2 e to the power of i theta end exponent over denominator 2 i end fraction equals fraction numerator minus i e to the power of i theta end exponent over denominator minus i squared end fraction equals minus i e to the power of i theta end exponent sum from k equals 1 to 6 of open parentheses sin space fraction numerator 2 πk over denominator 7 end fraction minus i space cos space fraction numerator 2 πk over denominator 7 end fraction close parentheses equals sum from k equals 1 to 6 of open parentheses minus i e to the power of fraction numerator i 2 πk over denominator 7 end fraction end exponent close parentheses equals minus i open parentheses e to the power of fraction numerator i 2 pi over denominator 7 end fraction end exponent plus e to the power of fraction numerator i 4 pi over denominator 7 end fraction end exponent plus e to the power of fraction numerator i 6 pi over denominator 7 end fraction end exponent plus e to the power of fraction numerator i 8 pi over denominator 7 end fraction end exponent plus e to the power of fraction numerator i 10 pi over denominator 7 end fraction end exponent plus e to the power of fraction numerator i 12 pi over denominator 7 end fraction end exponent close parentheses equals minus i open parentheses fraction numerator e open parentheses blank to the power of i begin display style fraction numerator 2 straight pi over denominator 7 end fraction end style end exponent close parentheses to the power of 6 minus 1 over denominator e to the power of begin display style fraction numerator i 2 straight pi over denominator 7 end fraction end style end exponent minus 1 end fraction close parentheses equals minus i open parentheses fraction numerator e to the power of blank to the power of i fraction numerator 12 straight pi over denominator 7 end fraction end exponent end exponent minus 1 over denominator e to the power of begin display style i fraction numerator 2 straight pi over denominator 7 end fraction end style end exponent minus 1 end fraction close parentheses

Answered by Prasenjit Paul | 4th Oct, 2014, 12:36: PM

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