Question

. Find the smallest positive number p for which the equation cos (p sin x) = sin (p cos x) has a
solution when x € (0,2π].
[IIT 1995)​

Answer 1

Answer:

π/2√2.

Step-by-step explanation:

cos(psinx)=sin(pcosx)

sin(π/2 – psinx)=sin(pcosx)

π/2 – psinx = pcosx

p(sinx + cosx)= π/2

p = π/2/(sinx + cosx)

So, to minimize p, sinx+cosx must be maximized.

sinx+cosx=√2sin(x+π/4) , which is maximized when sin(x+π/4)=1 at x=π/4.

p√2(sin(x+π/4))=π/2

p = π/2 / √2sin(π/2)

  = π/2√2.