Section 4.7 Inverse Trigonometric Functions

A brief review….. 1.If a function is one-to-one, the function has an inverse. 2.If the graph of a function passes the horizontal line test, then the function is one- to-one. 3.Some functions can be made to pass the horizontal line test by restricting their domains.

More… 4.If (a,b) is a point on the graph of f, then (b,a) is a point on the graph of f-inverse. 5.The domain of f-inverse is the range of f. 6.The range of f-inverse is the domain of f. 7.The graph of f-inverse is a reflection of the graph of f about the line y = x.

The inverse sine function Denoted by sin -1 The domain of y = sin x is restricted to y = sin -1 x means that sin y = x. sin -1 x is the angle, between –π/2 and π/2, inclusive, whose sine value is x.

The inverse cosine function Denoted by cos -1 The domain of y = cos x is restricted to y = cos -1 x means that cos y = x. cos -1 x is the angle, between 0 and π, inclusive, whose cosine value is x.

The inverse tangent function Denoted by tan -1 The domain of y = tan x is restricted to y = tan -1 x means that tan y = x. tan -1 x is the angle, between –π/2 and π/2, whose tangent value is x.

Evaluating inverse functions For exact values, use your table and/or your knowledge of the unit circle. For approximate values, use your calculator (be careful to watch your MODE).

Examples

Evaluating composite functions Composite functions come in two types: 1.The function is on the “inside”. 2.The inverse is on the “inside”. In either case, work from the “inside out”. Be sure to observe the restricted domains of the functions you are dealing with. Sometimes the function and inverse will “cancel” each other but, again, watch your restricted domains. For values not on the unit circle, draw a sketch and use right triangle trigonometry.

Examples

Weird Examples Use a right triangle to write the following expression as an algebraic expression: