Way to prove why the 4,3,2,1 pattern occurs in the unit circle coordinates?

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July 1st 2012, 09:29 PM
lamp23

Way to prove why the 4,3,2,1 pattern occurs in the unit circle coordinates?

Does anyone know of a good way of demonstrating why the $\frac{\sqrt4}{2},\frac{\sqrt3}{2},\frac{\sqrt2}{2} ,\frac{\sqrt1}{2}$ pattern occurs in the coordinates of the unit circle?
July 1st 2012, 09:40 PM
Prove It

Re: Way to prove why the 4,3,2,1 pattern occurs in the unit circle coordinates?

Quote:

Originally Posted by lamp23

Does anyone know of a good way of demonstrating why the $\frac{\sqrt4}{2},\frac{\sqrt3}{2},\frac{\sqrt2}{2} ,\frac{\sqrt1}{2}$ pattern occurs in the coordinates of the unit circle?

Yes, consider the unit circle and the two important right angle triangles, namely the isosceles right angle triangle, and the bisected equilateral triangle.
July 3rd 2012, 10:40 PM
lamp23

Re: Way to prove why the 4,3,2,1 pattern occurs in the unit circle coordinates?

I know how to prove each individual result but was wondering if there was any good way of demonstrating why it has the 1,2,3,4 pattern.
July 3rd 2012, 10:46 PM
richard1234

Re: Way to prove why the 4,3,2,1 pattern occurs in the unit circle coordinates?

Try squaring each number.

$\sin^2 0^{\circ} = \frac{0}{4}$

$\sin^2 30^{\circ} = \frac{1}{4}$

$\sin^2 45^{\circ} = \frac{2}{4}$

$\sin^2 60^{\circ} = \frac{3}{4}$

$\sin^2 90^{\circ} = \frac{4}{4}$
July 4th 2012, 05:47 AM
HallsofIvy

Re: Way to prove why the 4,3,2,1 pattern occurs in the unit circle coordinates?

I don't understand what you mean by that "pattern occurs in the coordinates of the unit circle". Every number from -1 to 1 occurs as a coordinate of a point on the unit circle. What "pattern" are you talking about?
July 4th 2012, 07:38 AM
richard1234

Re: Way to prove why the 4,3,2,1 pattern occurs in the unit circle coordinates?

He means, something to do with 30-60-90 and 45-45-90 triangles.
July 4th 2012, 09:13 AM
lamp23

Re: Way to prove why the 4,3,2,1 pattern occurs in the unit circle coordinates?

Quote:

Originally Posted by richard1234

He means, something to do with 30-60-90 and 45-45-90 triangles.

correct.
I was wondering if anything can be done similar to the spiral of theodorus

http://upload.wikimedia.org/wikipedi...odorus.svg.png
but obviously since we don't have the pattern occuring in the hypotenuse, I then wondered if it had something to do with 15 degree angle increases but realized that's not such a nice pattern since increasing from 0 to 30 requires two 15 degree increases which produces the same effect on $cos\theta$ as only 1 15 degree increase from 30 to 45.