6.6i Inverse Trigonometric Functions Inverse Sine Function Does g(x) = sin(x) have an inverse? What restriction would we need to make so that at least a piece of this function has an inverse? Given f (x) = sin(x); ) Find f (x) 2) Graph f (x) and f (x). ) Find the domain and range of f (x) and f (x). We define y = sin (x) or y = arcsin(x) to mean sin(y) = x AND π 2 y π 2 Note: Both the input and output of this function are real numbers, but it is sometimes helpful to think in terms of angles. sin(θ ) = x that is let θ = sin (x) or θ = arcsin(x) mean AND π 2 θ π 2
For example: sin π = 6 2 sin = π 2 6 sin( angle) = number sin ( number) = angle in π 2, π 2 Finding exact values of the inverse sine function for special inputs: (like: ) Ex: sin Set θ = sin and re-write according to the definition as In words: sin is the real number (or angle) in π 2, π whose sine (or y value on the 2 unit circle) is Ex: sin 2 2 Since y = sin (x) is a function, Ex: Ex: sin ( 2) Ex: sin 4 Compositions sin sin π = sin sin ( ( )) = sin sin 5π = sin sin 2 6 ( ( )) = 2
Think about it: sin sin 4π 5 = Inverse Cosine Function What restriction would we need to make so that at least a piece of this function has an inverse? Given f (x) = cos(x); ) Find f (x) 2) Graph f (x) and f (x). ) Find the domain and range of f (x) and f (x). We define y = cos (x) or y = arccos(x) to mean cos(y) = x AND 0 y π Note: Both the input and output of this function are real numbers, but it is sometimes helpful to think in terms of angles.
that is let θ = cos (x) or θ = arccos(x) mean cos(θ ) = x AND 0 θ π As before, it is helpful to think of the input/outputs as follows: cos( angle) = number cos ( number) = angle in [ 0, π] Finding exact values of the inverse sine function for special inputs: Ex: cos 2 In words: cos 2 unit circle) is Ex: cos 2 2 is the real number (or angle) in [ 0, π] whose cosine (or x value on the Ex: Ex: cos ( 0.2) Compositions cos cos π = cos cos 0.5 6 ( ( )) = cos cos 5π = cos cos.5 ( ( )) = Think about it: cos cos π = 2 4
Inverse Tangent Function What restriction would we need to make so that at least a piece of this function has an inverse? Given f (x) = tan(x); ) Find f (x) 2) Graph f (x) and f (x). ) Find the domain and range of f (x) and f (x). We define y = tan (x) or y = arctan(x) to mean tan(y) = x AND π 2 < y < π 2 As before, both the input and output of this function are real numbers, but it is sometimes helpful to think in terms of angles. 5
tan( angle) = number tan ( number) = angle in π 2, π Finding exact values of the inverse sine function for special inputs: Ex: tan 2 In words: tan is the real number (or angle) in [ 0, π] whose tangent Ex: tan ( ) Ex: Ex: tan ( 8) Compositions Similar to the case for cosine and sine, 6
The other inverse trig. functions The other inverses: f (x) = sec(x) Trig friendly restrictions: sec (2) = sec ( 2 ) = Calculus friendly restrictions: sec (2) = sec ( 2 ) = See the book for csc (x) and cot (x). You do not need to memorize these restrictions, but know how to find values for a given set of restrictions. 7
8 Mixed Compositions Find exact values: sin cos 5 tan sin 2 5 5 sin(2arctan()) cos tan 4 sin 2
6.6ii Derivatives of Inverse Trigonometric Functions Develop formula for the derivative of f (x) = sin (x) Similarly, we can derive the formulas for the other inverse trig function derivatives. d [ sin (x)] = d [ csc (x)] = d [ cos (x)] = d [ sec (x)] = d [ tan (x)] = d [ cot (x)] = Examples: 9
Integration. From the above derivative formulas, we gain the following antiderivative formulas: x 2 = = 2 + x x = x 2 Examples: / 2 0 x 2 x 9x 2 4 + x 2 Generalizing Formulas (a>0): = sin x +C a 2 x 2 a a 2 + x 2 = a tan x +C a = x x x 2 a 2 a sec +C a 0
ln(2 / ) ln(2) e x e 2x