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Convert the complex number i-1/cos60+isin60 into polar form.

Asked by Shubham Kumar | 9th Nov, 2014, 10:13: AM

Expert Answer:

$W e space n e e d space t o space f i n d space t h e space p o l a r space f o r m space o f 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 60 plus i sin 60 end fraction : L e t space a plus i b equals fraction numerator i minus 1 over denominator cos 60 plus i sin 60 end fraction rightwards double arrow a plus i b equals fraction numerator i minus 1 over denominator begin display style 1 half end style plus begin display style fraction numerator i square root of 3 over denominator 2 end fraction end style end fraction rightwards double arrow a plus i b equals fraction numerator 2 i minus 2 over denominator begin display style 1 plus i square root of 3 end style end fraction rightwards double arrow a plus i b equals fraction numerator 2 i minus 2 over denominator begin display style 1 plus i square root of 3 end style end fraction cross times fraction numerator 1 minus i square root of 3 over denominator 1 minus i square root of 3 end fraction rightwards double arrow a plus i b equals fraction numerator open parentheses 2 i minus 2 close parentheses cross times open parentheses 1 minus i square root of 3 close parentheses over denominator begin display style 1 plus 3 end style end fraction rightwards double arrow a plus i b equals fraction numerator 2 i plus 2 square root of 3 minus 2 plus 2 i square root of 3 over denominator begin display style 4 end style end fraction rightwards double arrow a plus i b equals fraction numerator square root of 3 minus 1 plus i open parentheses square root of 3 plus 1 close parentheses over denominator begin display style 2 end style end fraction a equals fraction numerator square root of 3 minus 1 over denominator 2 end fraction ; b equals fraction numerator square root of 3 plus 1 over denominator 2 end fraction square root of a squared plus b squared end root 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 fraction numerator square root of 3 plus 1 over denominator 2 end fraction over denominator fraction numerator square root of 3 minus 1 over denominator 2 end fraction end fraction rightwards double arrow 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 cross times fraction numerator square root of 3 plus 1 over denominator square root of 3 plus 1 end fraction rightwards double arrow theta equals tan to the power of minus 1 end exponent fraction numerator open parentheses square root of 3 plus 1 close parentheses squared over denominator 3 minus 1 end fraction rightwards double arrow theta equals tan to the power of minus 1 end exponent fraction numerator open parentheses 3 plus 1 plus 2 square root of 3 close parentheses over denominator 2 end fraction rightwards double arrow theta equals tan to the power of minus 1 end exponent fraction numerator open parentheses 4 plus 2 square root of 3 close parentheses over denominator 2 end fraction rightwards double arrow theta equals tan to the power of minus 1 end exponent open parentheses 2 plus square root of 3 close parentheses T h e r e f o r e comma space a plus i b equals r open square brackets cos theta plus i sin theta close square brackets rightwards double arrow space a plus i b 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 open parentheses 2 plus square root of 3 close parentheses$

Answered by Vimala Ramamurthy | 9th Nov, 2014, 01:59: PM

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