Same-function inverse compositions

Same-function inverse trigonometric compositions

Learn when a trigonometric function and its inverse undo one another and when the principal branch changes the result. This precalculus practice compares the two composition orders for sine, cosine and tangent. Domain checks come first, followed by the forward value and the permitted inverse output range.

Questions include exact compositions, angles outside a principal interval, a labeled folded-output graph and piecewise reasoning on a specified interval. Multiselect items ask which identities hold throughout their stated domains. Error analysis distinguishes a true restricted inverse relationship from a cancellation rule applied to every real angle.

Prerequisites are principal inverse sine, cosine and tangent, exact unit-circle values, coterminal angles and basic function composition. All angles are in radians. These tasks use the same trigonometric function on both sides of the composition; mixed-function expressions and general trigonometric equation solving belong to later topics.

The authored sequence moves from valid direct compositions through exact branch selection to graphical and interval-based justification. Worked explanations check both the forward ratio and the final angle range. Because a selected response is only provisional evidence of understanding, also explain the folding rule or sketch a principal branch without looking at the choices.