Pg. 385 Homework Pg. 395#13 – 41 odd, Graph the three inverse trig functions and label the domain and range of each. Memorization quiz through inverse trig functions on Monday!!

7.2 Inverse Trigonometric Functions Inverse Functions What is an inverse? How can you tell it is an inverse both algebraically and graphically? Will f(x) = x 2 – 4 have an inverse? – If so, prove it algebraically and graphically. Inverse sin x Consider y = sin x. Will it pass the HLT? Will it have an inverse? An inverse function can be defined as long as the domain of the original function lends itself to an inverse function. Consider y = sin x on the interval [-π/2, π/2]. Will it pass the HLT? Will it have an inverse function?

7.2 Inverse Trigonometric Functions Inverse Sine Function The inverse sine function, denoted y = sin -1 x or y = arcsin x is the function with a domain of [-1, 1] and a range of [-π/2, π/2] that satisfies the relation sin y = x. If f(x) = sin x and f -1 (x) = sin -1 x (f -1 ◦ f)(x) = x on [-π/2, π/2] and (f ◦ f -1 )(x) = x on [-1, 1] Inverse cos x Consider y = cos x. Will it pass the HLT? Will it have an inverse function? Consider y = cos x on the interval [0, π]. Will it pass the HLT? Will it have an inverse function?

7.2 Inverse Trigonometric Functions Inverse Cosine Function The inverse cosine function, denoted y = cos -1 x or y = arccos x is the function with a domain of [-1, 1] and a range of [0, π] that satisfies the relation cos y = x. If f(x) = cos x and f -1 (x) = cos -1 x (f -1 ◦ f)(x) = x on [0, π] and (f ◦ f -1 )(x) = x on [-1, 1] Inverse Tangent Function The inverse tangent function, denoted y = tan -1 x or y = arctan x is the function with a domain of (-∞, ∞) and a range of (-π/2, π/2) that satisfies the relation tan y = x. If f(x) = tan x and f -1 (x) = tan -1 x (f -1 ◦ f)(x) = x on (-π/2, π/2) and (f ◦ f -1 )(x) = x on (-∞, ∞)

7.2 Inverse Trigonometric Functions Find the Domain and Range. Graph. f(x) = sin -1 (2x) g(x) = sin -1 (⅓ x) h(x) = cos -1 (3x) k(x) = cos -1 (⅕ x)