tan A space a n d space tan B space left parenthesis tan A greater than tan B right parenthesis space a r e space t h e space r o o t s space o f space t h e space e q u a t i o n space 2 t squared minus t equals 3. space C a l c u l a t e left parenthesis w i t h o u t space u sin g space c a l c u l a t o r space a n d space t a b l e s right parenthesis comma space t h e space v a l u e s space o f space tan left parenthesis A plus B right parenthesis comma space tan left parenthesis A minus B right parenthesis space a n d space tan 2 A.

Asked by Olivia Adhikary | 4th Aug, 2014, 12:58: PM

Expert Answer:

C o n s i d e r space t h e space g i v e n space e q u a t i o n space 2 t squared minus t minus 3 equals 0 A l s o space g i v e n space t h a t space tan A space a n d space tan B space a r e space t h e space r o o t s space o f space t h e space a b o v e space q u a d r a t i c space p o l y n o m i a l comma space w h e r e comma space tan A greater than tan B. S u m space o f space t h e space r o o t s comma space tan A plus tan B equals minus b over a equals minus open parentheses fraction numerator minus 1 over denominator 2 end fraction close parentheses equals 1 half P r o d u c t space o f space t h e space r o o t s comma space tan A cross times tan B equals c over a equals fraction numerator minus 3 over denominator 2 end fraction W e space k n o w space t h a t comma tan open parentheses A plus B close parentheses equals fraction numerator tan A plus tan B over denominator 1 minus tan A cross times tan B end fraction rightwards double arrow tan open parentheses A plus B close parentheses equals fraction numerator 1 half over denominator 1 minus open parentheses fraction numerator minus 3 over denominator 2 end fraction close parentheses end fraction rightwards double arrow tan open parentheses A plus B close parentheses equals fraction numerator 1 half over denominator 1 plus open parentheses 3 over 2 close parentheses end fraction rightwards double arrow tan open parentheses A plus B close parentheses equals fraction numerator 1 half over denominator 5 over 2 end fraction rightwards double arrow tan open parentheses A plus B close parentheses equals 1 fifth W e space k n o w space t h a t space tan open parentheses A plus B close parentheses equals fraction numerator tan A minus tan B over denominator 1 plus tan A cross times tan B end fraction C o n s i d e r space t h e space i d e n t i t y open parentheses tan A minus tan B close parentheses squared equals open parentheses tan A plus tan B close parentheses squared minus 4 space tan A cross times tan B rightwards double arrow open parentheses tan A minus tan B close parentheses equals square root of open parentheses tan A plus tan B close parentheses squared minus 4 space tan A cross times tan B end root rightwards double arrow open parentheses tan A minus tan B close parentheses equals square root of open parentheses 1 half close parentheses squared minus 4 space open parentheses fraction numerator minus 3 over denominator 2 end fraction close parentheses end root rightwards double arrow open parentheses tan A minus tan B close parentheses equals square root of open parentheses 1 fourth close parentheses plus space open parentheses 24 over 4 close parentheses end root rightwards double arrow open parentheses tan A minus tan B close parentheses equals square root of open parentheses fraction numerator 1 plus 24 over denominator 4 end fraction close parentheses end root rightwards double arrow open parentheses tan A minus tan B close parentheses equals square root of open parentheses 25 over 4 close parentheses end root rightwards double arrow open parentheses tan A minus tan B close parentheses equals 5 over 2 C n s i d e r space t h e space g i v e n space q u a d r a t i c space e q u a t i o n comma space space 2 t squared minus t minus 3 equals 0 R e w r i t i n g space t h e space a b o v e space e q u a t i o n comma space w e space h a v e comma 2 t squared plus 2 t minus 3 t minus 3 equals 0 rightwards double arrow 2 t open parentheses t plus 1 close parentheses minus 3 open parentheses t plus 1 close parentheses equals 0 rightwards double arrow open parentheses 2 t minus 3 close parentheses open parentheses t plus 1 close parentheses equals 0 rightwards double arrow 2 t minus 3 equals 0 space o r space t plus 1 equals 0 rightwards double arrow t equals 3 over 2 space o r space t equals minus 1 S i n c e space tan A greater than tan B comma space w e space h a v e comma tan A equals t equals 3 over 2 W e space k n o w space t h a t space tan 2 A equals fraction numerator 2 tan A over denominator 1 minus tan squared A end fraction T h u s comma space w e space h a v e space tan 2 A equals fraction numerator 2 cross times 3 over 2 over denominator 1 minus open parentheses 3 over 2 close parentheses squared end fraction rightwards double arrow space tan 2 A equals fraction numerator 3 over denominator 1 minus begin display style 9 over 4 end style end fraction rightwards double arrow space tan 2 A equals fraction numerator 3 over denominator begin display style fraction numerator 4 minus 9 over denominator 4 end fraction end style end fraction rightwards double arrow space tan 2 A equals fraction numerator minus 12 over denominator 5 end fraction

Answered by Vimala Ramamurthy | 5th Aug, 2014, 09:09: AM

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