Find the smallest positive number p for which the equation `cos (p sin x) = sin (p cos x)` has a solution `x in [0,2pi]`.

Options

A. 2
B. 1
C. 3
D. none of these

Answer

A

Explanation

We have,
`"cos" (a "sin" x) = "sin" (a "cos" x)`
`rArr "cos" (a"sin" x) = "cos" ((pi)/(2)-a"cos" x)`
`rArr a"sin" x = (pi)/(2)-a"cos" x rArr "cos" x + "sin" x = (pi)/(2a)`
This equation will have a solution iff
`|(pi)/(2a)| le sqrt(2) " " [because |a "cos" x + b "sin" x | le sqrt(a^(2) + b^(2))]`
`rArr (pi)/(2sqrt(2)) le |a|`
`rArr a ge (pi)/(2sqrt(2)) " " [because a gt 0]`
`rArr a gt1 " " because 1 lt (pi)/(2sqrt(2)) lt 2]`