Question

Explain why domain restrictions are necessary to create inverse trigonometric functions rather than relations. two 3 sentences

Answer 1

Trigonometric functions are not one-to-one on their full domains (for example, sin(x) repeats the same values every 2π), so without restricting the domain an “inverse” would give multiple outputs for a single input and thus be a relation, not a function. By restricting the domain to an interval where the trig function is one-to-one (the principal branch, e.g. sin on [-π/2, π/2], cos on [0, π], tan on (-π/2, π/2]), each value corresponds to exactly one angle. That one-to-one restriction guarantees a well-defined inverse function.