Angle `alpha` is divided into two parts A and B such that A -B = x and tan A : B = P : q . The value of sin x is equal to

Options

A. `3sinalpha`
B. `(2sinalpha)//3`
C. `(sinalpha)//3`
D. `2sinalpha`

Answer

A

Explanation

Since , angle `alpha` is divided into two parts A and B.
`thereforealpha=A+B` . . . (1) and a = A -B (given) . . . (2)
On solving (1) and (2) we get ,
`A=(alpha+x)/(2),B(alpha-x)/(2)`
Now , consider
`(tanA)/(tanB)=(2)/(1)`
`rArr(tan((alpha+x)/(2)))/(tan((alpha-x)/(2)))=(2)/(1)`
`rArr(sin((alpha+x)/(2))cos((alpha-x)/(2)))/(cos((alpha-x)/(2))sin((alpha-x)/(2)))=(2)/(1)`
Multiply and divide by 2 ,
`rArr(2sin((alpha+x)/(2))cos((alpha-x)/(2)))/(2cos((alpha+x)/(2))sin((alpha+x)/(2)))=2`

`rArr(sinalpha+sinx)/(sinalpha-sinx)=2`
`rArrsinalpha+sinx=2sinalpha-2sinx`
`rArr3sinx=sinalpha`
`rArrsinx = (sinalpha)/(3)`