E x p r e s s space t h e space c o m p l e x space n u m b e r space i n space p o l a r space f o r m : fraction numerator i minus 1 over denominator c o s begin display style pi over 3 end style plus i space s i n begin display style pi over 3 end style end fraction

Asked by medha painuly | 16th Sep, 2014, 03:14: PM

Expert Answer:

C o n s i d e r space t h e space c o m p l e x space n u m b e r space fraction numerator i minus 1 over denominator cos begin display style straight pi over 3 end style plus i sin straight pi over 3 end fraction M u l t i p l y i n g space t h e space n u m e r a t o r space a n d space t h e space d e n o m i n a t o r space b y space i t s space c o n j u g a t e comma space w e space h a v e comma fraction numerator i minus 1 over denominator cos begin display style straight pi over 3 end style plus i sin straight pi over 3 end fraction equals fraction numerator i minus 1 over denominator cos begin display style straight pi over 3 end style plus i sin straight pi over 3 end fraction cross times fraction numerator cos straight pi over 3 minus i sin straight pi over 3 over denominator cos begin display style straight pi over 3 end style minus i sin straight pi over 3 end fraction equals fraction numerator open parentheses i minus 1 close parentheses open parentheses cos straight pi over 3 minus straight i sin straight pi over 3 close parentheses over denominator open parentheses cos straight pi over 3 plus i sin straight pi over 3 close parentheses open parentheses cos straight pi over 3 minus i sin straight pi over 3 close parentheses end fraction space space equals fraction numerator i cos begin display style straight pi over 3 end style minus i squared sin straight pi over 3 minus cos straight pi over 3 plus i sin straight pi over 3 over denominator cos squared open parentheses straight pi over 3 close parentheses minus i squared sin squared open parentheses straight pi over 3 close parentheses end fraction equals fraction numerator sin straight pi over 3 minus cos straight pi over 3 plus i sin straight pi over 3 plus i cos straight pi over 3 over denominator cos squared open parentheses straight pi over 3 close parentheses plus sin squared open parentheses straight pi over 3 close parentheses end fraction equals sin straight pi over 3 minus cos straight pi over 3 plus i open parentheses sin straight pi over 3 plus cos straight pi over 3 close parentheses equals fraction numerator square root of 3 over denominator 2 end fraction minus 1 half plus i open parentheses fraction numerator square root of 3 over denominator 2 end fraction plus 1 half close parentheses z equals fraction numerator square root of 3 minus 1 over denominator 2 end fraction plus i open parentheses fraction numerator square root of 3 plus 1 over denominator 2 end fraction close parentheses N o w space l e t space u s space c o n s i d e r comma space z equals a plus i b T h e n space t o space e x p r e s s space i n space p o l a r space f o r m comma space l e t space u s space f i n d space square root of a squared plus b squared end root r equals square root of open square brackets fraction numerator square root of 3 minus 1 over denominator 2 end fraction close square brackets squared plus open square brackets fraction numerator square root of 3 plus 1 over denominator 2 end fraction close square brackets squared end root equals square root of 2 theta equals tan to the power of minus 1 end exponent fraction numerator square root of 3 plus 1 over denominator square root of 3 minus 1 end fraction T h u s comma space z equals square root of 2 open square brackets cos theta plus i sin theta close square brackets comma space w h e r e space theta equals tan to the power of minus 1 end exponent fraction numerator square root of 3 plus 1 over denominator square root of 3 minus 1 end fraction space

Answered by Vimala Ramamurthy | 17th Sep, 2014, 11:11: AM