All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars
 perplexus dot info

 Trigonometry nest (Posted on 2005-08-12)
Which is greater, sin(cos(x)) or cos(sin(x))? Prove it!

 See The Solution Submitted by Federico Kereki Rating: 3.8333 (6 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
 Solution | Comment 8 of 10 |
` `
`Since the cos(x) and sin(x) are both periodic with a periodof 2 pi, we can restrict the domain to (-pi,pi].`
`Since cos(sin(x)) and sin(cos(x)) are both even, we canfurther restrict the domain to [0,pi].`
`Since cos(sin(x)) >= 0 and sin(cos(x)) < 0 in (pi/2,pi],we can further restrict our domain to [0,pi/2].`
`     f(x) = sin(x) + cos(x) >= 0`
`    f'(x) = cos(x) - sin(x) = 0 ==> x = pi/4`
`    f"(x) = -[sin(x) + cos(x)] < 0 at x = pi/4`
`     f(0) = f(pi/2) = 1`
`Therefore, in [0,pi/2]`
`  sin(x) + cos(x) <= sin(pi/4) + cos(pi/4) = sqrt(2) < pi/2`
`Since the sin(x) is strictly increasing in [0,pi/2],`
`  sin(cos(x)) < sin(pi/2 - sin(x)) = cos(sin(x))`
`Therefore, `
`  cos(sin(x)) > sin(cos(x))`
`for all real x.`
` `
` `

 Posted by Bractals on 2005-08-13 15:39:24

 Search: Search body:
Forums (0)