If `theta` and `phi` are acute angles satisfying `sin theta=1/2, cos phi=1/3` then `theta+phi=`

Options

A. `((pi)/(3),(pi)/(2))`
B. `((pi)/(2),(2pi)/(3))`
C. `((2pi)/(3),(5pi)/(6))`
D. `((5pi)/(6),pi)`

Answer

B

Explanation

As `sintheta=(1)/(2)` and `cosphi=(1)/(3)`
`impliestheta=(pi)/(6)" and "0lt(cosphi=(1)/(3))lt(1)/(2)["as", 0lt(1)/(3)lt(1)/(2)]`
`implies" "theta=(pi)/(6)`
and `cos^(-1)(0)gt phi gt cos^(-1)((1)/(2))`
[the sign changed as cos x is decreasing between `(0,(pi)/(2))`]
`impliestheta=(pi)/(6)" and "(pi)/(3)ltphilt(pi)/(2)`
`implies" "(pi)/(2)lttheta+philt(2pi)/(3)`
`:." "phi+thetain((pi)/(2),(2pi)/(3))`