Evaluating Compositions

Pre-Calculus · Axiom Academy

EXAMPLE Evaluating Compositions of Trig and Inverse Trig Functions Working a mixed composition with the right-triangle method Evaluate . Because the outer and inner functions are different (sine outside, inverse cosine inside), they don't simply cancel — so we set the inner angle as and build a right triangle. cos θ = adjacent / hypotenuse = 3/5, so the opposite side is √(5² − 3²) = √16 = 4. Nice work — you evaluated by turning an inverse trig value into a right triangle. Cancellation (matching functions): whenever . For instance , since is in [-1, 1] . Reverse order needs care: only when x is in the range . Because , you reduce with the reference angle , giving . Mixed composition (this problem): set equal to the inner inverse, draw the triangle from its ratio, find the missing side with a^2 + b^2 = c^2 , then read off the outer ratio. The right-triangle method turns any -style composition into a side-length you can simply read off.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.