Section 10.6: The Inverse Trigonometric Functions - Ostts.org

10.6 The Inverse Trigonometric Functions 831Theorem 10.29. Properties of the Arcsecant and Arccosecant Functions a• Properties of F (x) = arcsec(x)– Domain: {x : |x| ≥ 1} = (−∞, −1] ∪ [1, ∞)– Range: [ 0, π ) [ )2 ∪ π,3π2– as x → −∞, arcsec(x) → 3π −2 ; as x → ∞, arcsec(x) →π −2– arcsec(x) = t if and only if 0 ≤ t < π 2 or π ≤ t < 3π 2and sec(t) = x– arcsec(x) = arccos ( )1x for x ≥ 1 onlyb– sec (arcsec(x)) = x provided |x| ≥ 1– arcsec(sec(x)) = x provided 0 ≤ x < π 2 or π ≤ x < 3π 2• Properties of G(x) = arccsc(x)– Domain: {x : |x| ≥ 1} = (−∞, −1] ∪ [1, ∞)– Range: ( 0, π ] ( ]2 ∪ π,3π2– as x → −∞, arccsc(x) → π + ; as x → ∞, arccsc(x) → 0 +– arccsc(x) = t if and only if 0 < t ≤ π 2 or π < t ≤ 3π 2– arccsc(x) = arcsin ( )1x for x ≥ 1 onlyc– csc (arccsc(x)) = x provided |x| ≥ 1– arccsc(csc(x)) = x provided 0 < x ≤ π 2 or π < x ≤ 3π 2a . . . assuming the “Calculus Friendly” ranges are used.b Compare this with the similar result in Theorem 10.28.c Compare this with the similar result in Theorem 10.28.and csc(t) = xOur next example is a duplicate of Example 10.6.3. The interested reader is invited to compareand contrast the solution to each.Example 10.6.4.1. Find the exact values of the following.(a) arcsec(2) (b) arccsc(−2) (c) arcsec ( sec ( ))5π4(d) cot (arccsc (−3))2. Rewrite the following as algebraic expressions of x and state the domain on which the equivalenceis valid.(a) tan(arcsec(x))(b) cos(arccsc(4x))