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?
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?
Try squaring each number.
$\displaystyle \sin^2 0^{\circ} = \frac{0}{4}$
$\displaystyle \sin^2 30^{\circ} = \frac{1}{4}$
$\displaystyle \sin^2 45^{\circ} = \frac{2}{4}$
$\displaystyle \sin^2 60^{\circ} = \frac{3}{4}$
$\displaystyle \sin^2 90^{\circ} = \frac{4}{4}$
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?
correct.
I was wondering if anything can be done similar to the spiral of theodorus
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.