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 $\displaystyle \frac{\sqrt4}{2},\frac{\sqrt3}{2},\frac{\sqrt2}{2} ,\frac{\sqrt1}{2}$ pattern occurs in the coordinates of the unit circle?

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 $\displaystyle \frac{\sqrt4}{2},\frac{\sqrt3}{2},\frac{\sqrt2}{2} ,\frac{\sqrt1}{2}$ 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.

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.

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

Try squaring each number.

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

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

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

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

$\displaystyle \sin^2 90^{\circ} = \frac{4}{4}$

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?

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.

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 $\displaystyle cos\theta$ as only 1 15 degree increase from 30 to 45.