If a `DeltaABC`, the value of sinA+sinB+sinC is

Options

A. `4sin.(A)/(2)sin.(B)/(2)sin.(C)/(2)`
B. `4cos.(A)/(2)cos.(B)/(2)cos.(C)/(2)`
C. `4cos.(A)/(2)sin.(B)/(2)sin.(C)/(2)`
D. `4cos.(A)/(2)sin.(B)/(2)cos.(C)/(2)`

Answer

B

Explanation

`lnDeltaABC,A+B+C=180^(@)`
`impliessinA+sinB+sinC`
`=2sin.(A+B)/(2)cos.(A-B)/(2)+2sin.(C)/(2)cos.(C)/(2)`
`=2sin((pi)/(2)-(C)/(2))cos.(A-B)/(2)+2cos.(C)/(2)sin((pi)/(2)-bar(A+B)/(2))`
`=2cos.(C)/(2)cos.(A-B)/(2)+2cos.(C)/(2)cos.(A+B)/(2)`
`=2cos.(C)/(2)[cos.(A-B)/(2)+cos.(A+B)/(2)]`
`=2cos.(C)/(2)(2cos.(A)/(2)cos.(B)/(2))`
`=4cos.(A)/(2)cos.(B)/(2)cos.(C)/(2)`