The range of `theta` in the interval `(0,pi)` such that the points (3,5) and `("sin" theta, "cos" theta)` lie on the same side of the line x+y-1=0 is

Options

A. `(0,(pi)/(2))`
B. `(0,(pi)/(4))`
C. `((pi)/(4),(pi)/(2))`
D. none of these

Answer

A

Explanation

If the points (3,2) and `(cos theta , sin theta)` lie on the same side of `x+Y-1=0` , then
`3+2-1 and cos theta + sin theta-1` must be of the same sign.
`therefore cos theta +sin theta-1 lt 0`
`implies (1)/(sqrt(2))cos theta +(1)/(sqrt(2)) sin theta lt (1)/(sqrt(2))`
`implies cos theta" sin "(pi)/(4) + sin theta "cos" (pi)/(4) gt (1)/(sqrt(2))`
`implies sin (theta+(pi)/(4)) gt (1)/(sqrt(2))`
`implies (pi)/(4) lt theta +(pi)/(4) lt (3pi)/(4) implies 0 lt theta lt (pi)/(2) implies theta in(0,(pi)/(2))`