Graphing the sin(theta) and cos(theta) using the Unit Circle

The Unit Circle is a circle with a radius of 1, and which is centered at the origin (0,0).
Its equation is: x² + y² = 1.
Its equation is also: x = cos(theta), y = sin(theta).
HOMEWORK part one: PROVE  to yourself that x² + y² = 1
for every point  in the following diagram. 

For example,

(1,0) ==> x = 1 and y = 0 ==>
          x^2 = 1 and y^2 = 0 ==> 1 + 0 = 1    check

(√3/2, 1/2) ==> x = √3/2 and y = 1/2 ==>
          x^2 = 3/4 and y^2 = 1/4 ==> 3/4 + 1/4 = 1   check

(√2/2, √2/2) ==> x = √2/2 and y = √2/2 ==>
          x^2 = 2/4 and y^2 = 2/4 ==> 2/4 + 2/4 = 1  check

and so on for all the points around the circle.

HOMEWORK part two: PLAY with the following applets
for the sin(theta) http://www.ies.co.jp/math/java/trig/graphSinX/graphSinX.html
and cos(theta) http://www.ies.co.jp/math/java/trig/graphCosX/graphCosX.html

and PROVE that sin(theta) = y and cos(theta) = x, for all the points
in the unit circle. Note, Don't forget to rotate the cos(theta) applet
after you have drawn it !