1. 4.7 Inverse Trig Functions

2. Objective
Evaluate and graph the inverse sine, cosine and tangent function.
Evaluate the compositions of trig functions

3. Inverse Sine Function
Recall that for a function to have an inverse, it must be a one-to-one function and pass the Horizontal Line Test.
f(x) = sin x does not pass the Horizontal Line Test
and must be restricted to find its inverse. y x
y = sin x
Sin x has an inverse function on this interval.
Inverse Sine Function

4. Take a look at the piece of the graph in the red frame.
If we restrict the domain to the interval [-π/2, π/2]. The following properties hold:
On the interval [-π/2, π/2], the function y = sin x is increasing.
On the interval [-π/2, π/2],y = sin x takes on its full range of values, [-1, 1].
Lets zoom in and look at some key points in this section.

5. 3. On the interval [-π/2, π/2],y = sin x is one-to-one.

6. –1.5
1.5 – 

7. The range of y = arcsin x is [–/2 , /2].
The inverse sine function is defined by
y = arcsin x if and only if sin y = x.
Angle whose sine is x
The domain of y = arcsin x is [–1, 1].
The range of y = arcsin x is [–/2 , /2].
Example:
This is another way to write arcsin x.
Inverse Sine Function

8. Inverse Cosine Function
f(x) = cos x must be restricted to find its inverse.
The cosine function is decreasing and one-to-one on the interval [0, π] y x
y = cos x
Cos x has an inverse function on this interval.
Inverse Cosine Function

10. Inverse Cosine Function
The inverse cosine function is defined by
y = arccos x if and only if cos y = x.
Angle whose cosine is x
The domain of y = arccos x is [–1, 1].
The range of y = arccos x is [0 , ].
Example:
This is another way to write arccos x.
Inverse Cosine Function

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

13. Inverse Tangent Function
The inverse tangent function is defined by
y = arctan x if and only if tan y = x.
Angle whose tangent is x
The domain of y = arctan x is
The range of y = arctan x is (–/2 , /2).
Example:
This is another way to write arctan x.
Inverse Tangent Function

14. Composition of Functions
f(f –1(x)) = x and (f –1(f(x)) = x.
Inverse Properties:
If –1  x  1 and – /2  y  /2, then
sin(arcsin x) = x and arcsin(sin y) = y.
If –1  x  1 and 0  y  , then
cos(arccos x) = x and arccos(cos y) = y.
If x is a real number and –/2 < y < /2, then
tan(arctan x) = x and arctan(tan y) = y.
Example: tan(arctan 4) = 4
Composition of Functions

15. Composition of Functions
Example:
a. sin–1(sin (–/2)) = –/2
does not lie in the range of the arcsine function, –/2  y  /2. y x
However, it is coterminal with which does lie in the range of the arcsine function.
Composition of Functions

16. Example: Evaluating Composition of Functionsy 3 u 2
Example: Evaluating Composition of Functions