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c Math 151, Spring 2010, Benjamin Aurispa 4.6 Inverse Trigonometric Functions Remember that only one-to-one functions have inverses. So, in order to find the inverse functions for sine, cosine, and tangent, we must restrict their domains to intervals where they are one-to-one. To find the inverse sine function, we restrict the domain of sine to [−π/2, π/2]. We define the inverse sine function, sin−1 x by sin−1 x = y ↔ sin y = x. sin−1 x has domain and range . sin−1 x is the ANGLE in the interval [−π/2, π/2] whose sine is x. Remember that for inverse functions, we have “cancellation” laws. So, for the sine function sin(sin−1 x) = x for −1 ≤ x ≤ 1 sin−1 (sin x) = x for − π2 ≤ x ≤ π 2 Examples: sin−1 √ 3 2 sin(sin−1 arcsin(− 12 ) 9 10 ) sin−1 2 sin−1 (sin 4π 3 ) 1 arcsin(sin π5 ) c Math 151, Spring 2010, Benjamin Aurispa In order to have an inverse for cosine, we restrict the domain of cosine to the interval [0, π]. The inverse cosine function cos−1 is defined by cos−1 x = y ↔ cos y = x. cos−1 x has domain and range . cos−1 x is the ANGLE in the interval [0, π] whose cosine is x. Cancellation Laws. cos(cos−1 x) = x for −1 ≤ x ≤ 1 cos−1 (cos x) = x for 0 ≤ x ≤ π Examples: cos−1 (− arccos(0) cos(cos−1 12 11 ) √ 2 2 ) cos(cos−1 (− 71 )) cos−1 (cos 7π 6 ) 2 arccos(cos 3π 4 ) c Math 151, Spring 2010, Benjamin Aurispa In order to have an inverse for tangent, we restrict the domain of tangent to the interval (−π/2, π/2). The inverse tangent function tan−1 is defined by tan−1 x = y ↔ tan y = x. tan−1 x has domain lim arctan x = x→∞ and range . lim arctan x = x→−∞ tan−1 x is the ANGLE in the interval (−π/2, π/2) whose tangent is x. Cancellation Laws. tan(tan−1 x) = x for all x tan−1 (tan x) = x for − π2 < x < Examples: √ tan−1 (− 3) arctan(1) π )) tan−1 (tan(− 12 arctan(tan( 5π 4 )) 3 π 2 tan(arctan 1000) c Math 151, Spring 2010, Benjamin Aurispa Examples: Evaluate the following expressions. tan(sin−1 45 ) sin(2 cos−1 23 ) sin(tan−1 x) cot(cos−1 x) Derivatives of Inverse Trig Functions d 1 sin−1 x = √ dx 1 − x2 d 1 cos−1 x = − √ dx 1 − x2 d 1 tan−1 x = dx 1 + x2 d 1 sec−1 x = √ dx x x2 − 1 4 c Math 151, Spring 2010, Benjamin Aurispa Find the derivatives of the following functions. f (x) = arcsin(3x2 + ln x) y = x2 arctan(sec−1 x) √ g(t) = (cos−1 ( t2 + sin t))−2 5