Section 4.6 Inverse trigonometric functionsInverse sine functionarcsin x = sin−1x = y ⇔ sin y = xDOMAIN −1 ≤ x ≤ 1RANGE −π2≤ y ≤π2CANCELLATION EQUATIONSsin−1(sin x) = x for −π2≤ x ≤π2sin(sin−1x) = x for − 1 ≤ x ≤ 1Example 1. Find1. sin−1(0.5)2. sin−1(sin 1)3. sin(2 sin−135)4. arcsin(sin5π4)1Example 2. Simplify tan(sin−1x)(sin−1x)′=1√1 − x2Inverse cosine functionarccos x = cos−1x = y ⇔ cos y = xDOMAIN −1 ≤ x ≤ 1RANGE 0 ≤ y ≤ πCANCELLATION EQUATIONScos−1(cos x) = x for 0 ≤ x ≤ πcos(cos−1x) = x for − 1 ≤ x ≤ 1(cos−1x)′= −1√1 − x2Inverse tangent functionarctan x = tan−1x = y ⇔ tan y = xDOMAIN −∞ ≤ x ≤ ∞RANGE −π2< y <π22CANCELLATION EQUATIONStan−1(tan x) = x for −π2≤ x ≤π2tan(tan−1x) = x for − ∞ ≤ x ≤ ∞limx→−∞tan−1x = −π2limx→∞tan−1x =π2(tan−1x)′=11 + x2Inverse cotangent functionarccotx = cot−1x = y ⇔ cot y = xDOMAIN −∞ ≤ x ≤ ∞RANGE 0 < y < πCANCELLATION EQUATIONScot−1(cot x) = x for 0 ≤ x ≤ πcot(cot−1x) = x for − ∞ ≤ x ≤ ∞limx→−∞cot−1x = 0limx→∞cot−1x = π(cot−1x)′= −11 + x2Other inverse trigonometric functionscsc−1x = y ⇔ csc y = xDOMAIN |x| ≥ 13RANGE y ∈0,π2 ∪π,3π2 (csc−1x)′= −1x√x2− 1sec−1x = y ⇔ sec y = xDOMAIN |x| ≥ 1RANGE y ∈0,π2∪π,3π2(sec−1x)′=1x√x2− 1Example 3. Differentiate each function:1. f(x) = sin−1(2x − 1)2. g(x) = x cos−1x −√1 − x23. h(x) = sin−1(tan−1x)4. u(t) =
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