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If in triangle A B C ,/C=45^0 then find ...

If in triangle `A B C ,/_C=45^0` then find the range of the values of `sin^2A+sin^2Bdot`

Text Solution

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`E=sin^(2)A+sin^(2)B`
`=sin^(2)A+sin^(2)(135^(@)-A)`
`=sin^(2)A+cos^(2)(45^(@)-A)`
`=1+sin^(2)A-sin^(2)(45^(@)-A)`
`=1+sin(2A-45^(@))sin 45^(@)`
`=1+(1)/sqrt(2)sin(2A-45^(@))`
Now `0ltA lt135^(@)`
`rArr 0lt2Alt270^(@)`
`rArr -45^(@)lt2A-45^(@)lt225^(@)`
`rArr -(1)/sqrt(2)ltsin(2A-45^(@))le1`
`rArr -(1)/(2)lt(1)/sqrt(2)sin(2A-45^(@))le(1)/sqrt(2)`
`rArr(1)/(2)lt1+(1)/sqrt(2)sin(2A-45^(@))le1+(1)/sqrt(2)`
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