1.

sin A + 2 sin 2 A + sin 3 Ais equal to which of the following ? 1.4sin2Acos^(2)((A)/(2)) 2.2 sin2A("sin"(A)/(2)+"cos"(A)/(2))^(2)3.8sin A cos A cos ^(2)((A)/(2))Select the correct answer using the code given below :

Answer»

1 and 2 only
2 and 3 only
1 and 3only
1 , 2 and 3

Solution :LET`A=30^(@)`
`rArrsinA+2sin2A=sin3A=sin30^(@)+2sin60^(@)+sin90^(@)`
`=(1)/(2)+(2sqrt(3))/(2)+1=(2sqrt(3)+3)/(2)`
`(because2cos^(2)A=1+cos2A)`
Now , `4sin2Acos^(2)((A)/(2))=2sin2A[1+cosA]`
`=2sin60^(@)[1+COS30^(@)]=(2sqrt(3)+3)/(2)`
Also ,`sin2A=2sinAcosA&SIN^(2)A+cos^(2)A=1`
`2sin2A["sin"(A)/(2)+"cos"(A)/(2)]^(2)`
`=2sin2A["sin"^(2)(A)/(2)"cos"^(2)(A)/(2)+2"sin"(A)/(2)"cos"(A)/(2)]`
`=2sin2A[1+sinA]=2sin60^(@)[1+sin30^(@)]`
`=(3sqrt(3))/(2)`
& 8 sin A cos A`cos^(2)((A)/(2))`
`=4sinAcosA[1+cosA]`
`=4sin30^(@)cos30^(@)[1+cos30^(@)]`
`=(2sqrt(3)+3)/(2)`


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