Chapter 4 Trigonometric Functions 1

4.7 Inverse Trigonometric Functions Objectives:  Evaluate inverse sine functions.  Evaluate other inverse trigonometric functions.  Evaluate compositions of trigonometric functions. 2

Inverse Functions  Recall that a function and its inverse reflect over the line y = x.  What must be true for a function to have an inverse?  It must be one-to-one, that is, it must pass the horizontal line test. 3

More Inverse Functions  Are sine, cosine, and tangent one-to-one?  If not, what must we do so that these functions will have inverse functions?  Hint: Consider y = x 2.  We must restrict the domain of the original function. 4

Sine and Its Inverse  f(x) = sin x does not pass the Horizontal Line Test  It must be restricted to find its inverse. 5 y x y = sin x Sin x has an inverse function on this interval.

Inverse Sine Function  The inverse sine function is defined by y = arcsin x if and only if sin y = x.  The domain of y = arcsin x is [–1, 1].  The range of y = arcsin x is _____________.  Why are the domain and range defined this way? 6 Angle whose sine is x

What Does “arcsin” Mean?  In an inverse function, the x -values and the y -values are switched.  So, arcsin x means the angle (or arc) whose sin is x.  Notation for inverse sine  arcsin x  sin -1 x 7

Examples  If possible, find the exact value. 8

Graphing Arcsine  Create a table for sin y = x for –π/2 ≤ y ≤ π/2.  Graph x on horizontal axis and y on vertical axis. 9 y–π/2–π/4–π/60π/6π/4π/2 x

Graph of Arcsine 10

Inverse Cosine Function  f(x) = cos x must be restricted to find its inverse. 11 Cos x has an inverse function on this interval. y x y = cos x

Inverse Cosine Function  The inverse cosine function is defined by y = arccos x if and only if cos y = x.  The domain of y = arccos x is [–1, 1].  The range of y = arccos x is [0, π].  Notation for inverse cosine:  arccos x or cos -1 x 12 Angle whose cosine is x

Examples  If possible, find the exact value 13

Graphing Arccos  Create a table for cos y = x for 0 ≤ y ≤ π.  Graph x on horizontal axis and y on vertical axis. 14 y0π/6π/3π/22π/35π/6π x

Graph of Arccos 15

Inverse Tangent Function  f(x) = tan x must be restricted to find its inverse. 16 Tan x has an inverse function on this interval. y x y = tan x

Inverse Tangent Function  The inverse tangent function is defined by y = arctan x if and only if tan y = x.  The domain of y = arctan x is (–∞, ∞).  The range of y = arctan x is (–π/2, π/2).  Notation for inverse tangent:  arctan x or tan -1 x 17 Angle whose tangent is x

Examples  If possible, find the exact value 18

Graph of Arctan 19

Examples  Evaluate using your calculator. (What mode should the calculator be in?) 20

Summary 21

Composition of Functions  Given the restrictions specified in the previous slide, we have the following properties of inverse trig functions. 22

Examples  If possible, find the exact value. 23

24 Example x y 3 2 u

 Find the exact value of. 25

Homework 4.7  Worksheet