A periodic function needs a restricted branch before it can have an inverse.
Sine and cosine are not one-to-one on the entire real line. Their principal branches select the displayed intervals. Arctangent uses (−π/2,π/2) and accepts all real inputs; this explorer shows inputs from −1 to 1. Applying the original trigonometric function recovers the input.
Inverse trigonometric functions are not reciprocal trigonometric functions.