Mrs. Scott
Honors Trig/Pre-Calc
Name:_________________________________
Date:__________________________________
4.7 – Inverse Trig Functions
Objectives:
1. Evaluate inverse sine functions.
2. Evaluate other inverse trig functions.
3. Evaluate compositions of trig functions.
Recall:
What does it mean for a function to have an inverse?
How do you determine if a function has an inverse?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Draw a sketch of the sine function here:
Consider the sine function. It is not one-to-one. We can restrict the domain of the function so that we can have
an inverse function.
What do you think we should restrict the domain to?
Once we restrict the domain, we have the inverse sine function.
This is denoted by _______________ or _______________.
Sketch a graph of the inverse sine function:
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Definition of Inverse Sine Function
The inverse sine function is defined by:
y = arcsin x iff sin y = x.
What is the domain of the restricted
function sin x = y?
What is the domain of the inverse
function sin y = x?
What is the range of the restricted
function sin x = y?
What is the domain of the inverse
function sin y = x?
*******Remember: “the arcsine of x is the angle (or number) whose sine is x.”*******
Where does the term “arcsine” come from?
Arcsin x means:
Example 1:
If possible, find the exact value.
1
2
a. arcsin(  )
b. sin 1
2
2
c. sin 1 2
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The Inverse Cosine Function
Draw a sketch of the cosine function here:
Consider the cosine function. It is not one-to-one. We can restrict the domain of the function so that we can
have an inverse function.
What do you think we should restrict the domain to?
Once we restrict the domain, we have the inverse cosine function.
This is denoted by _______________ or _______________.
The inverse cosine function is defined by:
y = arccos x iff cos y = x.
Sketch the graph of the inverse cosine function:
Definition of Inverse Cosine Function
The inverse cosine function is defined by:
y = arccos x iff cos y = x.
What is the domain of the restricted
function cos x = y?
What is the domain of the inverse
function cos y = x?
What is the range of the restricted
function cos x = y?
What is the domain of the inverse
function cos y = x?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The Inverse Tangent Function
Draw a sketch of the tangent function here:
Consider the tangent function. It is not one-to-one. We can restrict the domain of the function so that we can
have an inverse function.
What do you think we should restrict the domain to?
Once we restrict the domain, we have the inverse tangent function.
The inverse tangent function is defined by: y = arctan x iff tan y = x.
Sketch a graph of the inverse tangent function:
What is the range?
Definition of Inverse Tangent Function
The inverse tangent function is defined by: y = arctan x iff tan y = x.
What is the domain of the restricted
function tan x = y?
What is the domain of the inverse
function tan y = x?
What is the range of the restricted
function tan x = y?
What is the domain of the inverse
function tan y = x?
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Example 2:
a. arccos(
Find the exact value.
2
)
2
b. cos 1 ( 1)
c. arctan 0
d. tan 1 ( 1)
Composition of Functions
What do you recall about the composition of inverse functions?
This is no different when working with trig functions.
Inverse Properties
If 1  x  1 and 


 y  , then _______________________________________________________.
2
2
If 1  x  1 and 0  y   , then _______________________________________________________.


 y  , then ______________________________________________________.
2
2
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Example 3: If possible, find the exact value.
If x is a real number and 
a. tan[arctan(-5)]
b. arcsin( sin
5
)
3
c. cos( cos 1  )
Example 4:
Find the exact value.
2
a. tan(arccos )
3
3
b. cos[arcsin ( )]
5
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Example 5: Write each of the following as an algebraic expression in x.
a. sin(arcos 3x), 0  x 
1
3
HW: pg. 316 #1-4all, 7, 31-45odd, 51, 54
b. cot(arcos 3x), 0  x 
1
3