Cauchy-Schwarz Inequality

**Jukodan** · June 26th 2010, 11:47 AM

I am supposed to prove that for all real numbers $a,b, \theta$ that
$(acos\theta+bsin\theta)^2\leq a^2+b^2$
I've only gotten up to this point which is where I am not sure as to how to appply the cauchy schwarz inequality,

$(acos\theta+bsin\theta)^2=a^2+b^2-(asin\theta+bcos\theta)^2$

Am I on the right track?

**dwsmith** · June 26th 2010, 12:19 PM

$<acosx+bsinx,acosx+bsinx>=<acosx,acosx+bsinx>+<bsi nx,acosx+bsinx>$

$=<acosx,acosx>+2<acosx,bsinx>+<bsinx,bsinx>=.....$

Also, I am assuming inner product is defined as such:

$\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)g(x)dx$ because if this is the case, everything works out just fine.

**Jukodan** · June 26th 2010, 12:41 PM

It does not specify the inner product. I have posted all the information that was given to me.

**dwsmith** · June 26th 2010, 12:43 PM

Originally Posted by Jukodan

It does not specify the inner product. I have posted all the information that was given to me.

Try finishing the problem with that inner product in mind and you will have your problem completed.

**Plato** · June 26th 2010, 12:52 PM

Originally Posted by Jukodan

I am supposed to prove that for all real numbers $a,b, \theta$ that
$(acos\theta+bsin\theta)^2\leq a^2+b^2$

$\left( {a\sin (t) - b\cos (t)} \right)^2 = a^2 \sin ^2 (t) - 2ab\sin (t)\cos (t) + b^2 \cos (t)$ Note the minus.
So we get $\left( {a\cos (t) + b\sin (t)} \right)^2 = a^2 + b^2 - \left( {a\sin (t) - b\cos (t)} \right)^2$
Notice that $a^2 + b^2 - \left( {a\sin (t) - b\cos (t)} \right)^2 \le a^2+b^2$

**dwsmith** · June 26th 2010, 01:32 PM

$<acosx+bsinx,acosx+bsinx>=<acosx,acosx+bsinx>+<bsi nx,acosx+bsinx>$

$=<acosx,acosx>+2<acosx,bsinx>+<bsinx,bsinx>=.....$

Spoiler:

**Jukodan** · June 26th 2010, 02:04 PM

Ah-ha!

**Opalg** · June 27th 2010, 12:28 PM

Originally Posted by Jukodan

I am supposed to prove that for all real numbers $a,b, \theta$ that
$(acos\theta+bsin\theta)^2\leq a^2+b^2$
I've only gotten up to this point which is where I am not sure as to how to appply the cauchy schwarz inequality,

$(acos\theta+bsin\theta)^2=a^2+b^2-(asin\theta+bcos\theta)^2$

Am I on the right track?

The C–S inequality says that $x_1y_1+x_2y_2\leqslant (x_1^2+x_2^2)^{1/2}(y_1^2+y_2^2)^{1/2}$ . Apply that with $x_1=a$ , $x_2=b$ , $y_1=\cos\theta$ , $y_2 = \sin\theta$ . Then square both sides.

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