Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Section 5.6 Inverse Trigonometric Functions: Differentiation * None of the six basic trigonometric functions has an inverse function ( they are not one-to-one ) , unless you fix / restrict their domains. y = sin x y = sin x not one-to-one one-to-one with restricted domain Domain: [ −π 2, π 2] Range: [ −1,1] Domain of y = sin x ⇒ Range of y = arcsin x Range of y = sin x ⇒ Domain of y = arcsin x Domain: [ −1,1] reflect restricted sine function about y = x = y arcsin = x sin −1 x = y arcsin = x iff sin y x Domain: [ −1,1] Range: [ −π 2, π 2] Range: [ −π 2, π 2] y = cos x y = cos x one-to-one with restricted domain not one-to-one Domain: [ 0, π ] Range: [ −1,1] Domain: [ −1,1] reflect restricted cosine function about y = x = y arccos = x cos −1 x = y arccos = x iff cos y x Domain: [ −1,1] Range: [ 0, π ] Range: [ 0, π ] y = tan x vertical asymptote y = tan x not one-to-one one-to-one with restricted domain = y arctan = x tan −1 x Domain: ( −π 2, π 2 ) Range: ( −∞, ∞ ) Domain: ( −∞, ∞ ) Range: ( −π 2, π 2 ) reflect restricted tangent function about y = x = y arctan = x iff tan y x Domain: ( −∞, ∞ ) Range: ( −π 2, π 2 ) horizontal asymptote y = cot x not one-to-one y = cot x Domain: ( 0, π ) one-to-one with restricted domain reflect restricted cotangent function about y = x Range: ( −∞, ∞ ) −1 = y arc = cot x cot x = y arc = cot x iff cot y x Domain: ( −∞, ∞ ) Range: ( 0, π ) Domain: ( −∞, ∞ ) Range: ( 0, π ) y = sec x not one-to-one y = sec x one-to-one with restricted domain Domain: ( 0, π ) x ≠ π 2 Range: ( −∞, −1] [1, ∞ ) y arc sec x sec −1 x = = reflect restricted secant function about y = x Domain: ( −∞, −1] [1, ∞ ) Range: ( 0, π ) y ≠ π 2 = y arc = sec x iff sec y x Domain: ( −∞, −1] [1, ∞ ) Range: ( 0, π ) y ≠ π 2 y = csc x not one-to-one y = csc x one-to-one with restricted domain Domain: ( −π 2, π 2 ) x ≠ 0 Range: ( −∞, −1] [1, ∞ ) = y arccsc = x csc −1 x reflect restricted cosecant function about y = x Domain: ( −∞, −1] [1, ∞ ) Range: ( −π 2, π 2 ) y ≠ 0 = y arc = csc x iff csc y x Domain: ( −∞, −1] [1, ∞ ) Range: ( −π 2, π 2 ) y ≠ 0 Definitions of Inverse Trigonometric Functions Range Domain Function y arcsin = x iff sin y x y arccos = x iff cos y x y arctan = x iff tan y x −1 ≤ x ≤ 1 −1 ≤ x ≤ 1 −∞ < x < ∞ arc = cot x iff cot y x −∞ < x < ∞ y y arc = sec x iff sec y x y arc = csc x iff csc y x x ≥1 x ≥1 −π 2 ≤ y ≤ π 2 0≤ y ≤π −π 2 < y < π 2 0< y <π 0 ≤ y ≤π, y ≠ π 2 −π 2 ≤ y ≤ π 2, y ≠ 0 Inverse Relationship Between Trigonometric Functions and Inverse Trigonometric Functions If − 1 ≤ x ≤ 1 and −π 2 ≤ y ≤ π 2, then = sin ( arcsin x ) x and = arcsin ( sin y ) y If − 1 ≤ x ≤ 1 and 0 ≤ y ≤ π , then = cos ( arccos x ) x and = arccos ( cos y ) y = = If −π 2 < y < π 2, then tan ( arctan x ) x and arctan ( tan y ) y If 0 < y < π , then = cot ( arc cot x ) x and = arc cot ( cot y ) y If x ≥ 1 and 0 ≤ y < π 2 or π 2 < y ≤ π , then = sec ( arc sec x ) x and = arc sec ( sec y ) y If x ≥ 1 and −π 2 ≤ y < 0 or 0 < y < π 2, then = csc ( arc csc x ) x and = arc csc ( csc y ) y Derivatives of Inverse Trigonometric Functions d u′ arcsin u = [ ] dx 1− u2 d −u ′ arccos u = [ ] dx 1− u2 d u′ [arctan u ] = dx 1+ u2 −u ′ d [arc cot u ] = 1+ u2 dx d u′ [arc sec u ] = dx u u2 −1 d −u ′ [arc csc u ] = dx u u2 −1