Compositions of Trig and Inverse Trig

Pre-Calculus · Axiom Academy

LESSON Compositions of Trig and Inverse Trig When do a trig function and its inverse cancel — and when does the cancellation quietly fail? 1. Inverse First: a Clean Round Trip — On Its Domain Put the inverse on the inside . Feeding x to produces an angle whose sine is x ; applying then reads that sine straight back. The trip closes perfectly — as long as x is a legal sine value , i.e. . Push x past 1 and doesn't even exist, so there is nothing to cancel. works for all real x (tangent takes every value) 2. Trig First: the Cancellation Can Lie Now put the trig function on the inside . only when already lives in arcsine's output range . Start outside it and throws away which copy of the angle you had — so hands back the one angle in range with the same sine , the reference angle, not your original . If , then . The round trip is honest. . You get the in-range angle sharing the same sine instead. Only on — cosine's inverse uses a different branch than sine's. Only on , the open branch tangent is built on. , and so does . Since sits outside , returns the in-range twin . Reduce first: subtract from to find the reference angle, keep its sign for . 3. Mixed Compositions: Let a Triangle Do the Work What about — a different trig outside than inverse inside? It won't collapse to x , but it isn't stuck. Name the inner angle , so . Draw the right triangle with opposite =x , adjacent =1 ; the hypotenuse is forced to , and now every trig ratio of is just a side ratio.

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