Inverse Trig Summary

Pre-Calculus · Axiom Academy

SUMMARY Inverse Trigonometric Functions A recap of the inverse trig functions, their restricted ranges, evaluating and composing them, and solving trigonometric equations. Sine, cosine, and tangent aren't one-to-one, so each inverse is defined on a restricted range — the principal values: on , on , on . An inverse undoes its function only on that range : always, but only when already lives in the range. For a mixed composition like , draw a right triangle (or use ) to get an algebraic answer: . Solving gives infinitely many answers: find every solution in one cycle, then add the period to get the general solution (or for tangent). (also written ) answers "which angle in the right half of the circle has sine x ?" Inputs run over [-1,1] ; outputs are the principal values in . Watch out for: the output is never above or below . (or ) returns the angle in the upper half of the circle whose cosine is x . Inputs again run over [-1,1] , but the range is — so the output is never negative. Watch out for: a negative input gives an obtuse angle (in ), not a negative one. (or ) accepts any real number and returns an angle strictly between and . The endpoints are excluded because tangent has vertical asymptotes there. Watch out for: as , but never reaches it.

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