# Proving an inequality

1. Aug 14, 2011

### Cosmossos

Last edited by a moderator: May 5, 2017
2. Aug 14, 2011

### SammyS

Staff Emeritus
Show what you have tried, and where you got stuck.

That way it will be easier for use to give the appropriate help.

3. Aug 14, 2011

### sir_manning

Don't forget about the reverse triangle inequality: |x - y| >= ||x| - |y||

4. Aug 15, 2011

### Cosmossos

Last edited by a moderator: May 5, 2017
5. Aug 15, 2011

### sir_manning

It will be easier to break this into two problems. First prove that $\left| sin z \right| \leq \frac{e^{y} + e^{-y}}{2}$ , then prove that $\left| sin z \right| \geq \frac{e^{\left| y \right|} + e^{- \left| y \right|}}{2}$.

Also, use $\left| sin z \right| = \frac{e^{i(x+iy)} - e^{-i(x+iy)}}{2i}$ .

6. Aug 16, 2011

### Cosmossos

That's what i did. can you please look at my answer? isn't it correct?
thank you.

7. Aug 16, 2011

### sir_manning

Sorry, I see that you did write $\left| sin z \right| = \frac{e^{i(x+iy)} - e^{-i(x+iy)}}{2i}$. However, I don't understand how you came up with your answer: where did the absolute value signs in $\left| \frac{e^{y} + e^{-y}}{2} \right|$ emerge from? You can't just insert them. And how did you re-arrange the inequality? Was there a typo in your original statement of the problem? In any case, your answer doesn't prove the inequality, because I cannot see its validity just by looking at it. With these types of problems, you really need to break it down to something like $-e^{-y} \leq e^{-y} \; \Rightarrow \; -1 \leq 1$, which we can all agree is true. Also, in proofs you *need* to show your steps, and you always should here anyways so we can help you out.

Alright, let's try doing this one part at a time. First, prove that:

$\left| sin z \right| \leq \frac{e^{y} + e^{-y}}{2}$, or

$\left| \frac{e^{i(x+iy)} - e^{-i(x+iy)}}{2i} \right| \leq \frac{e^{y} + e^{-y}}{2}$. Cancel the 2's, multiply by i/i and rearrange exponentials on the left,

$\left| -i e^{ix} e^{-y} + i e^{-ix} e^y \right| \leq e^{y} + e^{-y}$ ....Now try applying the triangle inequality to this. After proving this, a similar approach is used for $\left| sin z \right| \geq \frac{e^{\left| y \right|} + e^{- \left| y \right|}}{2}$

Last edited: Aug 16, 2011