The reason why '''Asin[sin(n)]''' may not equal '''n''':
Each of the trigonometric functions is periodic in the real part of its argument, running through all its values twice in each interval of <big>2<big><math> \pi </math></big></big>.
'''Sine''' and '''cosecant''' begin their period at <big>2<big><math> \pi </math></big>k − <big><math> \pi </math></big>/2</big> (where <big>k</big> is an integer), finish it at <big>2<big><math> \pi </math></big>k + <big><math> \pi </math></big>/2</big>, and then reverse themselves over <big>2<big><math> \pi </math></big>k + <big><math> \pi </math></big>/2</big> ───► <big>2<big><math> \pi </math></big>k + 3<big><math> \pi </math></big>/2</big>.
'''Cosine''' and '''secant''' begin their period at <big>2<big><math> \pi </math></big>k</big>, finish it at <big>2<big><math> \pi </math></big>k + <big><math> \pi </math></big></big>, and then reverse themselves over <big>2<big><math> \pi </math></big>k + <big><math> \pi </math></big></big> ───► <big>2<big><math> \pi </math></big>k + 2<big><math> \pi </math></big></big>.
'''Tangent''' begins its period at <big>2<big><math> \pi </math></big>k − <big><math> \pi </math></big>/2</big>, finishes it at <big>2<big><math> \pi </math></big>k + <big><math> \pi </math></big>/2</big>, and then repeats it (forward) over <big>2<big><math> \pi </math></big>k + <big><math> \pi </math></big>/2</big> ───► <big>2<big><math> \pi </math></big>k + 3<big><math> \pi </math></big>/2</big>.
'''Cotangent''' begins its period at <big>2<big><math> \pi </math></big>k</big>, finishes it at <big>2<big><math> \pi </math></big>k + <big><math> \pi </math></big></big>, and then repeats it (forward) over <big>2<big><math> \pi </math></big>k + <big><math> \pi </math></big></big> ───► <big>2<big><math> \pi </math></big>k + 2<big><math> \pi </math></big></big>.
<br>The above text is from the Wikipedia webpage: http://en.wikipedia.org/wiki/Inverse_trigonometric_functions
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