sin A + 2 sin 2 A + sin 3 A is 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 :

Options

A. 1 and 2 only
B. 2 and 3 only
C. 1 and 3 only
D. 1 , 2 and 3

Answer

C

Explanation

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)`