solve the equation for `x , 5^(1/2)+5^(1/2 + log_5 sinx) = 15^(1/2 + log_15 cosx)`

Options

A. 50
B. 100
C. 200
D. 400

Answer

A

Explanation

`5^((1)/(2))+5^((1)/(2)+log_(5)(sin x))=15^((1)/(2)+log_(5)cos x)`
`rArr 5^((1)/(2))5^((1)/(2)).5^(log_(5)(sin x))=15^(1//2).15^(log_(15)cos x)`
`rArr 1+sin x = sqrt(3)cos x`
`rArr (sqrt(3))/(2)cos x-(sin x)/(2)=(1)/(2)`
`rArr cos(x + (pi)/(6))=cos.(pi)/(3)`
`rArr x + (pi)/(6)=2n pi pm(pi)/(3), n in Z`
`rArr x = 2n pi-(pi)/(2), 2n pi + (pi)/(6), n in Z`
But we must have sin `x, cos x gt 0`
`therefore x = 2n pi + pi//6, n in Z`