First, forget about right triangles. That definition of "sine" and "cosine" is only true for angles between 0 and 90 degrees and doesn't work here.
A more general definition of sine and cosine is this: Start at the point with coordinates (1,0). Now, for t>= 0, measure, counterclockwise, around the unit circle a distance t: the coordinates of the point you end at are, by definition, (cos(t), sin(t)) (If t is negative, then measure clockwise).
Now, that's the UNIT circle: it has radius 1 and diameter 2- therefore circumference 2π. To find sin(t+ 2π), you would measure around the circle a distance t, then an additional 2π. Since that additional distance is exactly the circumference of the circle, it takes you right back to the original point so that sin(t+ 2π) and sin(t) give exactly the same thing: the y coordinate of the point (and cos(t+ 2π) and sin(t) are both equal to the x coordinate).
Notice that t, in that definition, isn't an angle at all! It is a measure along the circumference of the circle. However, calculators are not designed by mathematicians, they are designed by engineers and engineers think of sine and cosine in terms of angles! The "radian" is defined so that the angle, in radians, is exactly the same as the distance around the unit circle.
That's the reason you calculator is not giving the "correct" answer is that your calculator is in "degree" mode rather than "radian" mode. Some calculators have a "degree, radian, grad" or "d,r,g" key for changing from one to the other. Some have a "mode" key that allows you to change a variety of things including "radian" or "degree" mode.
You should understand that, basically, the only time you use "degrees" in when you are working with actual angles that are measured in degrees. When you are working with the trig functions as actual "functions", you always use "radians".
thank you
