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# the maximum value of the expression 1/sin^2a+3sinacosa+5cos^2a

8 years ago

multiply the numerator and the denominator by 2

numerator will become 2               &

the denominator             =   2sin^2a+6sinacosa+10cos^2a

taking a (sin^2a + cos^2a) aside  we get

=   sin^2a +6sinacosa +9cos^2a +sin^2a + cos^2a

=   (sina + 3 cosa)^2  + sin^2a +cos^2a

=  (sina + 3 cosa)^2  + 1

now the expression becomes                      2/( (sina + 3 cosa)^2  + 1  )

for the maximum value of the expression  the denominator must be minimum

and for denominator to be minimum , (sina + 3 cosa)^2  must be minimum

the minimum value of , (sina + 3 cosa)^2  can be zero   when   tana = -3

therefore the maximum value of the expression        2/( (sina + 3 cosa)^2  + 1  ) =  2/(  0 +1 )

= 2

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