Evaluating Inverse Trig Functions

Pre-Calculus · Axiom Academy

EXAMPLE Evaluating Inverse Trig Functions Reading an exact value off the unit circle, then choosing the one angle that lives in the function's restricted range. Evaluate three inverse trig expressions: , , and . For each, the real work is the same: find an angle with the right ratio, then keep the one answer that falls inside that inverse function's restricted range. The three principal ranges. Every answer below must land on the colored bar for its function — that is what rules out a same-ratio angle like −π/3 for arccos (it falls off the blue bar). Nice work. You evaluated all three inverse trig functions by asking the same question each time — "what angle in the restricted range has this ratio?" — and used the range to throw out the impostor angle. The method: find an angle with the right sine, cosine, or tangent, then keep only the one inside the function's restricted range. The ranges differ: uses , uses the open , but uses — so cosine answers never go negative. The classic trap: , not — both have the right cosine value, but is outside . The restricted range is what makes an inverse trig function a function at all — it is the single rule that turns infinitely many same-ratio angles into one answer.

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