Come up with a trig identity for $\displaystyle \frac{\sqrt{2}}{2}$ valid for ALL angles.

You are not allowed to use the number 2.

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- Dec 9th 2009, 06:32 AMrainertrig identity for sqrt(2)/2
Come up with a trig identity for $\displaystyle \frac{\sqrt{2}}{2}$ valid for ALL angles.

You are not allowed to use the number 2. - Dec 9th 2009, 08:04 PMWilmer
http://www.mathhelpforum.com/math-he...qrt-2-2-a.html

Isosceles right triangle with hypotenuse = 1 has equal sides = sqrt(2)/2;

you knew that? - Dec 10th 2009, 08:46 AMrainer
- Dec 10th 2009, 12:13 PMDefunkt
There is no such "identity". Why are you trying to find something that doesn't exist?

$\displaystyle \frac{\sqrt{2}}{2}sin^2(\theta) + \frac{\sqrt{2}}{2}cos^2(\theta) = \frac{\sqrt{2}}{2}$ - Dec 10th 2009, 01:19 PMWilmer
Methinks you better read them Google sites, Rainer (Cool)

- Dec 11th 2009, 06:13 AMrainer
- Dec 11th 2009, 08:21 AMBruno J.
It would not be a very big deal, but you'd win an argument.

- Dec 11th 2009, 01:45 PMDefunkt
Well then, please be so kind as to show us your findings!

- Dec 12th 2009, 09:22 AMrainer

I don't know man. What's it worth to you? I feel like I should get something out of this. Like a special plaque or something. "Discoverer of the sqrt(2)/2 trig identity, heretofore unknown to humans." At least a t-shirt. Or I suppose I would be satisfied with a check for $100. - Dec 12th 2009, 10:28 AMe^(i*pi)
- Dec 12th 2009, 10:33 AMrainer
Ok, $25.

- Dec 12th 2009, 10:46 AMe^(i*pi)
- Dec 12th 2009, 11:25 AMrainer
Just kidding.

Haven't you guys ever noticed how $\displaystyle r=\frac{1}{\sqrt{1+\cos^2x}}$ gives you a vertical unit ellipse of semi-minor axis $\displaystyle b=\frac{\sqrt{2}}{2}$ ?

This is basically the common polar equation that is usually given for the ellipse. Where does the b come from?

If you actually derive the equation yourself from scratch using trigonometry, you find out:

$\displaystyle r=\sqrt{\frac{b^2\sec^2\theta}{b^2+\tan^2\theta}}$

When you substitute $\displaystyle \sin{\theta}$ for $\displaystyle b$ the equation simplifies to: $\displaystyle r=\frac{1}{\sqrt{1+\cos^2\theta}}$

This in iteself is intriguing. An astuter mathematician will be able to derive the identity in question just from this info. But in my case I was not sure how to manipulate the b of the vertical ellipse, so I rotated it 90 degrees...

$\displaystyle r=\frac{1}{\sqrt{1+\cos^2(\theta+\frac{\pi}{2})}}= \frac{1}{\sqrt{1+\sin^2\theta}}$

This gives the same ellipse with the same b, but horizontally oriented. Next I just set the two equations equal to each other and solved for b (where b of course = $\displaystyle \frac{\sqrt{2}}{2}$):

$\displaystyle r=\sqrt{\frac{b^2\sec^2\theta}{b^2+\tan^2\theta}}= \frac{1}{\sqrt{1+\sin^2\theta}}$

(where $\displaystyle b=\frac{\sqrt{2}}{2}$)

drumroll please...

$\displaystyle b=\frac{\sqrt{2}}{2}=\sqrt{\frac{\tan^2\theta}{\si n^2\theta(\tan^2\theta+\sec^2\theta+1)}}=\frac{1}{ \sqrt{\cos^2\theta(\tan^2\theta+\sec^2\theta+1)}}$

Square both sides and you get an identity for $\displaystyle \frac{1}{2}$ too. If anyone finds some genius application for this I'd like to hear about it.

Now, can anyone derive the hyperbolic function correlate? - Dec 12th 2009, 04:04 PMDefunkt
Well...

$\displaystyle \frac{1}{\sqrt{\cos^2\theta(\tan^2\theta+\sec^2\th eta+1)}} = \frac{1}{ \sqrt{\frac{cos^2\theta \cdot \sin^2\theta}{cos^2\theta} + \frac{cos^2\theta}{cos^2\theta} + cos^2\theta}}$ $\displaystyle = \frac{1}{\sqrt{sin^2\theta + 1 + cos^2\theta}} = \frac{1}{\sqrt{2}}$

So there's nothing really new here. :) - Dec 12th 2009, 04:39 PMBruno J.
Drumroll.... oops! (Surprised)