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CBSE Class 11-science Answered

sin A/2*sinB/2*sinC/2 less or equal than 8
Asked by aahnik.mohanty | 20 May, 2019, 12:56: PM
answered-by-expert Expert Answer
A comma space B comma space a n d space C space a r e space t h e space a n g l e s space o f space a space t r i a n g l e space A B C L e t space f equals sin A over 2 sin B over 2 sin C over 2 equals sin A over 2 sin B over 2 cos fraction numerator A plus B over denominator 2 end fraction space.... space S i n c e space A plus B plus C equals straight pi fraction numerator partial differential f over denominator partial differential A end fraction equals sin B over 2 open square brackets cos A over 2 cos fraction numerator A plus B over denominator 2 end fraction minus sin A over 2 sin fraction numerator A plus B over denominator 2 end fraction close square brackets equals sin B over 2 cos open parentheses fraction numerator 2 A plus B over denominator 2 end fraction close parentheses fraction numerator partial differential f over denominator partial differential B end fraction equals sin A over 2 cos open parentheses fraction numerator A plus 2 B over denominator 2 end fraction close parentheses fraction numerator partial differential squared f over denominator partial differential B partial differential A end fraction equals 1 half cos left parenthesis A plus B right parenthesis fraction numerator partial differential squared f over denominator partial differential A partial differential B end fraction equals 1 half cos left parenthesis A plus B right parenthesis fraction numerator partial differential squared f over denominator partial differential A squared end fraction equals negative sin B over 2 sin open parentheses fraction numerator 2 A plus B over denominator 2 end fraction close parentheses fraction numerator partial differential squared f over denominator partial differential B squared end fraction equals negative sin A over 2 sin open parentheses fraction numerator A plus 2 B over denominator 2 end fraction close parentheses N o w comma space fraction numerator partial differential f over denominator partial differential A end fraction equals 0 space a n d space fraction numerator partial differential f over denominator partial differential B end fraction equals 0 rightwards double arrow sin B over 2 cos open parentheses fraction numerator 2 A plus B over denominator 2 end fraction close parentheses equals 0 space a n d space sin A over 2 cos open parentheses fraction numerator A plus 2 B over denominator 2 end fraction close parentheses space equals 0 rightwards double arrow cos open parentheses fraction numerator 2 A plus B over denominator 2 end fraction close parentheses equals 0 space a n d space cos open parentheses fraction numerator A plus 2 B over denominator 2 end fraction close parentheses space equals 0 space... space sin c e space sin B over 2 not equal to 0 space a n d space sin A over 2 not equal to 0 rightwards double arrow 2 A plus B equals straight pi space and space straight A plus 2 straight B equals straight pi rightwards double arrow straight A equals straight pi over 3 space and space straight B equals straight pi over 3 rightwards double arrow straight C equals straight pi over 3 Therefore comma space straight f space will space be space maximum space when space all space three space angles space are space equal space to space straight pi over 3 rightwards double arrow sin straight A over 2 sin straight B over 2 sin straight C over 2 less or equal than sin open parentheses fraction numerator straight pi over 3 over denominator 2 end fraction close parentheses sin open parentheses fraction numerator straight pi over 3 over denominator 2 end fraction close parentheses sin open parentheses fraction numerator straight pi over 3 over denominator 2 end fraction close parentheses equals sin straight pi over 6 sin straight pi over 6 sin straight pi over 6 equals 1 half cross times 1 half cross times 1 half equals 1 over 8 Hence comma space sin straight A over 2 sin straight B over 2 sin straight C over 2 less or equal than 1 over 8
Answered by Renu Varma | 21 May, 2019, 09:55: AM

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