Question

For all `theta` in [0,pi]`, show that `cos (sin theta) ge sin (cos theta)`.

Answer 1

We have, ` cos theta + sin theta = sqrt2 ( (1)/(sqrt2) cos theta + (1)/(sqrt2) sin theta ) `
`" " = sqrt2(sin ""(pi)/(4)*cos theta + cos ""(pi)/(4) *sin theta ) `
` " " =sqrt2 sin ( (pi)/(4) + theta )`
` rArr cos theta + sin theta le sqrt 2 lt (pi)/(2)`
`" " [ as, sqrt2 = 1.4141, (pi)/(2) = 1.57`(approx)`]`
` rArr " " cos theta + sin theta lt (pi)/(2)`
Since, ` " " cos theta lt (pi)/(2) - sin theta `
` rArr sin (cos theta ) lt sin ((pi)/(2) - sin theta ) `
`rArr " " sin (cos theta ) lt cos (sin theta )`
` rArr " " cos (sin theta ) gt sin (cos theta )`