Hey! Are you really just in 8th grade?
You're really really smart !! 
I had no clue about trig in the 8th grade...
I wanted to add to your explanation about sine.
To find the sin of the angle that a line from the origin to a specific point makes with the x-axis...a point like this (4, 7) :
1. Find the distance from the point to the origin. This is the radius.
2. Divide the y coordinate, in this case 7, by that distance.
Dividing by the radius scales the length down to unit circle measures.
For this one, use the Pythagorean theorem or distance formula (same thing really) to find that the radius = \(\sqrt{65}\)
And the sine of the angle that a line from the origin to this point makes with the x-axis = \(\frac{7}{\sqrt{65}}\)
If the radius is already = 1, then just get the y-coordinate of the point! :)
Also... to find the sin if an angle is given, exactly what you said works but I think you want to start at the right-most point on the circle. On a clock, that's 3:00. In a unit circle, that is the point (1, 0). Then go counter-clockwise. 
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