# Thread: good trignometric inequality

1. ## good trignometric inequality

prove that $x^{2} \sin x + x \cos x + x^{2} + \frac{1}{2}>0$ for any real number $x$

2. ## Re: good trignometric inequality

I started by trying to show the expression as the sum of squares:
$x^2\sin x + x \cos x + x^2 + \frac{1}{2}$
$= x^2\sin x + x \cos x + x^{2}(\sin^{2} x + \cos^{2} x)+\frac{1}{4}+ \frac{1}{4}$
$=(x^{2} \cos^{2} x + x \cos x + \frac{1}{4})+x^{2} \sin^{2} x + x^{2} \sin x + \frac{1}{4}$
$=(x \cos x + \frac{1}{2})^2 + x^2 \sin^2 x + x^2 \sin x + \frac{1}{4}$

any suggestion on how to go about from here......???

Advanced thanks for any help or suggestions....

3. ## Re: good trignometric inequality

Originally Posted by earthboy
prove that $x^{2} \sin x + x \cos x + x^{2} + \frac{1}{2}>0$ for any real number $x$
Wow this is a fun inequality. This is what I did, it is similar to yours.

$0\leq\frac{(x\sin(x)+x)^2}{2}=\frac{1}{2}(x^2\sin^ 2(x) + 2x^2\sin(x)+x^2)$

$0\leq\frac{(x\cos(x)+1)^2}{2}=\frac{1}{2}(x^2\cos^ 2(x) + 2x\cos(x)+1)$

Adding the two gives

$0\leq \frac{1}{2}(x^2(\sin^2(x)+\cos^2(x))+2(x^2\sin(x)+ x\cos(x))+x^2+1)$

$0\leq x^{2} \sin x + x \cos x + x^{2} + \frac{1}{2}$

I'm not sure about if the strict inequality case is true. If it is, the adjustment would be minor (just show that if one of the expressions is zero if the other one is strictly positive).

4. ## Re: good trignometric inequality

Originally Posted by earthboy
prove that $x^{2} \sin x + x \cos x + x^{2} + \frac{1}{2}>0$ for any real number $x$
I did it pretty much term by term.

Note that
$x^2~sin(x) + x^2 = x^2(sin(x) + 1)$
As sin(x) has a minimum of -1, this expression is always 0 or positive.

Note that
$x~cos(x) \geq 0$ since both x and cos(x) are odd functions.

And of course 1/2 is always positive.

Add them all up and you get your inequality.

-Dan

5. ## Re: good trignometric inequality

great proof! Gusbob!

Originally Posted by topsquark
Note that
$x~cos(x) \geq 0$ since both x and cos(x) are odd functions.

-Dan
Thanks Dan!, but can you or anybody please explain how this part is true ???
As $x$ is any real number, what about if $x=120$ , then $x \cos x = -60$ which is $\leq 0$.

anyways..advanced thanks as usual!!!

6. ## Re: good trignometric inequality

great proof! Gusbob!

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7. ## Re: good trignometric inequality

Originally Posted by earthboy
great proof! Gusbob!

Thanks Dan!, but can you or anybody please explain how this part is true ???
As $x$ is any real number, what about if $x=120$ , then $x \cos x = -60$ which is $\leq 0$.

anyways..advanced thanks as usual!!!
Well, that's ummm....because I screwed up. I had mentioned that cos(x) is an odd function. Actually it's an even function. Thus, please ignore my earlier post.

-Dan