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CHAPTER 1.7
CHAPTER 1 TRIGONOMETRY
PART 7 – Inverse Trigonometric Functions
TRIGONOMETRY MATHEMATICS CONTENT STANDARDS:
 8.0 – Students know the definitions of the inverse trigonometric functions and
can graph the functions.
 9.0 - Students compute, by hand, the values of the trigonometric functions and
the inverse trigonometric functions at various standard points.
OBJECTIVE(S):
 Students will learn the definition of the inverse sine, cosine, and
tangent function.
 Students will learn how to graph and evaluate the inverse
trigonometric functions.
 Students will learn how to evaluate composite trigonometric functions.
Inverse Sine Function
Recall from Algebra II that for a function to have an inverse, it must pass the
______________________ ________ _________ (it must be one-to-one). You can see
that y  sin x does not pass the test because different values of x yield the same y-value.
y  sin x
Do you see a problem?
y
1
x
-1
sin x has an inverse function on this interval


However, if you restrict the domain to the interval   x  (corresponding to the
2
2
black portion of the graph), the following properties hold:
CHAPTER 1.7
  
1. On the interval  ,  , the function y  sin x is __________________.
 2 2
  
2. On the interval  ,  , y  sin x takes on its full range of values,______________.
 2 2
  
3. On the interval  ,  , y  sin x passes the _____________ _______ _________.
 2 2


So, on the restricted domain   x  , y  sin x has a unique inverse called the
2
2
inverse sine function. It is denoted by
_______________________ or
________________.
The notation sin 1 x is consistent with the inverse function notation f 1 x  . The arcsin x
notation (read as “the arcsine of x”) comes from the association of a central angle with its
intercepted arc length on a unit circle. So, arcsin x means the angle (or arc) whose sine is
x. Both notations, arcsin x and sin 1 x are commonly used in mathematic, so remember
1
that sin 1 x denotes the inverse since function rather than
. The values of arcsin x lie
sin x


in the interval   arcsin x  . The graph of y = arcsin x is show below:
2
2
Definition of Inverse Sine Function
The inverse sine function is defined by
y  arcsin x
if and only if sin y  x
where, 1  x  1 and 
  
is  ,  .
 2 2

2
 y

2
. The domain of y  arcsin x is  1,1 and the range
CHAPTER 1.7
EXAMPLE 1: Evaluating the Inverse Sine Function
If possible, find the exact value.
 1
a.) arcsin   
 2
1


because sin______ =  for   y  , it follows that
2
2
2
 1
arcsin    =
 2
b.) sin 1
1
Angle whose sine is  .
2
3
2
because sin______ =
sin 1
3
=
2
3


for   y  , it follows that
2
2
2
Angle whose sine is
3
.
2
c.) sin 1 2
It is not possible to evaluate y  sin 1 x when x = ___ because there is ____ ________
whose sine is 2. Remember that the domain of the inverse sine function is_______.
CHAPTER 1.7
EXAMPLE 2: Graphing the Arcsine Function
Sketch a graph of y  arcsin x .
By definition, the equations y  arcsin x and sin y  x are equivalent for
___________________________. So, their graphs are the same. From the interval
_______________, you can assign values to y in the second equation to make a table of
values. Then plot the points and draw a smooth curve through the points.
x  sin y
y


2


4


6
0

6

4

2
y
-1
1 x
CHAPTER 1.7
Other Inverse Trigonometric Functions
The Cosine function is decreasing on the interval 0  x   .
y
x
Consequently, on this interval the cosine function has an inverse function – the inverse
cosine function – denoted by
y  arccos x
or
y  cos 1 x
Similarly, you can define an inverse tangent function by restricting the domain of
  
y  tan x to the interval   ,  .
 2 2
Definition of Inverse Trigonometric Functions
Function
y  arcsin x if and only if sin y  x
y  arccos x if and only if cos y  x
y  arctan x if and only if tan y  x
Domain
Range
CHAPTER 1.7
y
y
x
y
x
x
Domain:
Domain:
Domain:
Range:
Range:
Range:
EXAMPLE 3: Evaluating Inverse Trigonometric Functions
Find the exact value.
2
2
Because cos________ = ________, and _________ lies in ____________, it
follows that
2
2
arccos
=
Angle whose cosine is
.
2
2
a.) arccos
b.) cos 1  1
Because cos________ = ________, and _________ lies in ____________, it
follows that
Angle whose cosine is -1.
cos 1  1 =
c.) arctan 0
Because tan________ = ________, and _________ lies in ____________, it
follows that
arctan 0 =
Angle whose tangent is 0.
d.) tan 1  1
Because tan________ = ________, and _________ lies in ____________, it
follows that
tan 1  1 =
Angle whose tangent is -1.
CHAPTER 1.7
Evaluate
- By definition, when dealing with inverse trigonometric functions, answers must
be in radians.
 3

1.) sin 1 
2.) arccos 1
3.) tan 1  1

 2 

3

5.) cos 1  

2


 1
4.) arcsin   
 2
6.) arctan
 3
EXAMPLE 4: Calculators and Inverse Trigonometric Functions
Use a calculator to approximate the value (if possible).
Function
a.) arctan  8.45
Mode
Answer
b.) sin 1 0.2447
c.) arccos 2
because the domain of the inverse cosine function is ________
DAY 1
Compositions of Functions
Recall from Algebra II that for all x in the domains of f and f 1 , inverse functions have
the properties


f f 1 x   x
and
f 1  f x   x
CHAPTER 1.7
Inverse Properties of Trigonometric Functions


If 1  x  1 and   y  , then
2
2
sin arcsin x  x
arcsin sin y   y
and
If 1  x  1 and 0  y   , then
cosarccos x  x
If x is a real number and 
tan arctan x  x
arccoscos y   y
and

2
 y

2
, then
arctan tan y   y
and
Keep in mind that these inverse properties do not apply for arbitrary values of x and y.
For instance,
 3 
arcsin  sin
  ___________________ = ________  ___________
2 

In other words, the property
arcsin sin y   y
Is not valid for values of y outside the interval _________________.
EXAMPLE 5: Using Inverse Properties
If possible, find the exact value.
a.) tanarctan  5
Because _______ lies in the domain of arctan x , the inverse property applies, and you
have
tanarctan  5
=
_____________.
CHAPTER 1.7
 5 
b.) arcsin  sin

3 

In this case, __________ does not lie within the range of the arcsine function,
___________________________. However, _______ is co terminal with
=
which does lie in the range of the arcsine function, and you have
 5 
arcsin  sin

3 


c.) cos cos 1 
=
=

The expression ______________________ is not defined because __________________
is not defined. Remember that the domain of the inverse cosine function is ___________.
EXAMPLE 6: Evaluating Compositions of Functions

2

 3 
Find the exact value of (a) tan  arccos  and (b) cos arcsin    .
3

 5 

2
a.) If you let u  arccos , then cos u  _________. Because cos u is positive, u is a first3
quadrant angle.
y
x
2

tan  arccos  = _______________________ = _______________ = __________
3

CHAPTER 1.7
 3
b.) If you let u  arcsin    , then sin u  __________. Because sin u is negative, u is a
 5
fourth-quadrant angle.
y
x

 3 
cos arcsin    = _______________________ = _______________ = __________
 5 

CHAPTER 1.7
EXAMPLE 7: Some Problems from Calculus
Write each of the following as an algebraic expression in x.
a.) sin arccos 3x , 0  x 
1
3
b.) cot arccos 3x , 0  x 
1
3
If you let u = ____________________, then cos u  ________. Because
cos u  _________ = ____________
you can sketch a right triangle with acute angle u. From this triangle, you can easily
convert each expression to algebraic form.
a.) sin arccos 3x
=
b.) cot arccos 3x
=
=
=
=
=
=
7.) Evaluate each expression:
a. sin arcsin .7
11 

b. arctan  tan

6 

CHAPTER 1.7

 3 
8.) Find the exact value of sec arctan    .
 5 

DAY 2
CHAPTER 1.7
9.) Write an algebraic expression that is equivalent to the expression.
a. sin arccos x , let u  arccos x .
b. tan arcsin x , let u  arcsin x .
DAY 3