1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Inequality involving theta and sin(theta)

  1. Jun 3, 2009 #1
    1. The problem statement, all variables and given/known data
    I'm trying to prove the following inequality for [tex]0\leq \theta\leq\frac{\pi}{2}[/tex]

    [tex] \frac{2}{\pi} \theta \leq\sin\theta[/tex]

    2. Relevant equations

    [tex] 0 \leq \theta\leq\frac{\pi}{2} [/tex]

    [tex] 0\leq \frac{2}{\pi}\theta\leq 1 [/tex]

    3. The attempt at a solution
    I've looked at the limit

    [tex] \lim_{\theta\to 0}\frac{\sin\theta}{\theta}=1 [/tex]

    I've also looked at other inequalities involving [tex]\sin\theta[/tex]


    I've also tried to take derivatives to see how fast [tex]\sin\theta[/tex] grows in comparison to [tex]\theta[/tex], but I have not managed to prove the inequality.
  2. jcsd
  3. Jun 3, 2009 #2


    Staff: Mentor

    That limit won't do you much good, since it is for x near zero.

    The graphs of y = 2x/[itex]\pi[/itex] and y = sin(x), you'll see that the sine curve is above the line, and that the two intersect at the origin and at ([itex]\pi[/itex], 1), and at no other points on the interval you're interested in. This isn't a proof, though, but you should be able to convey what the graph shows through calculus.

    You should be able to establish your inequality by showing that the graph of y = sin(x) is concave down on the interval (0, [itex]\pi[/itex]), meaning that the graph of y = sin(x) will be above the graph of y = 2x/[itex]\pi[/itex] except at the endpoints of your interval.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook