4.7 Inverse Trigonometric Functions
Inverse Sine Function
The inverse of y=sinx is not a function because it does not pass the horizontal line test. But an interval of the function could be used.If the domain was restricted to the interval -π/2 ≤ x ≤ π/2, then there are several properties held.
So with the restricted domain -π/2 ≤ x ≤ π/2, y=sinx has a unique inverse function called the inverse sine function denoted y=arcsinx y=sin⁻¹x |
Definition of Inverse Sine Function
The inverse sine function is defined by y=arcsinx
if and only if siny=x where -1≤x≤1 and -π/2 ≤ y ≤ π/2.
if and only if siny=x where -1≤x≤1 and -π/2 ≤ y ≤ π/2.
Evaluating the Inverse Sine Function
Graphing the Arcsine Function
Other Inverse Trigonometric Functions
Function Domain Range
y = arcsin x if and only if sin y = x -1 ≤ x ≤ 1 -π/2 ≤ y ≤ π/2
y = arccos x if and only if cos y = x -1 ≤ x ≤ 1 0≤ y ≤ π
y = arctan x if and only if tan y = x -∞ < x < ∞ -π/2 < y < π/2
y = arcsin x if and only if sin y = x -1 ≤ x ≤ 1 -π/2 ≤ y ≤ π/2
y = arccos x if and only if cos y = x -1 ≤ x ≤ 1 0≤ y ≤ π
y = arctan x if and only if tan y = x -∞ < x < ∞ -π/2 < y < π/2
Evaluating Inverse Trigonometric Functions
Parent Functions
Library of Parent Function: f(x) = arccos x
Compositions of Functions
Inverse PropertiesIf -1 ≤ x ≤ 1 and -π/2 ≤ y ≤ π/2, then
sin(arcsin x) = x and arcsin(sin y) = y
If -1 ≤ x ≤ 1 and 0 ≤ y≤ π, then
cos(arccos x) = x and arccos(cos y) = y
If x is a real number and -π/2 < y < π/2, then
tan(arctan x) = x and arctan(tan y) = y
The inverse properties do not apply for arbitrary values of x and y. The property is not valid for values of y outside the intervals.
sin(arcsin x) = x and arcsin(sin y) = y
If -1 ≤ x ≤ 1 and 0 ≤ y≤ π, then
cos(arccos x) = x and arccos(cos y) = y
If x is a real number and -π/2 < y < π/2, then
tan(arctan x) = x and arctan(tan y) = y
The inverse properties do not apply for arbitrary values of x and y. The property is not valid for values of y outside the intervals.
Using Inverse Properties
Evaluating Compositions of Functions