First of all it is NOT 'pie' . 'Pie' is the foodstuff that you eat e,g, apple pie, meat pie.
The Mathemtical 'pi' is spelled as 'pi' and is from the Classical Greek alphabet, and represent the small /lower case letter 'p'.
To answer your question.
Set your calculator in RADIAN Mode.
Sin(pi/3) sqrt(3) /2 = 0.8660254....
NB You will have an incorrect answer if your calculator is set in 'DEGREE' Mode.
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1/3
sin(2*pi) - not pie - is the same as sin(0) = 0
Sin(pi/6) = 1/2 is a true statement [not pie].
There is no reason at all. For most angles sin plus cos do not equal one.
Because the slope of the curve of sin(x) is cos(x). Or, equivalently, the limit of sin(x) over x tends to cos(x) as x tends to zero.