7.4 Inverse Trigonometric Functions I
OBJECTIVE 1: Understanding and Finding the Exact and Approximate Values of the Inverse Sine
Function
y  sin x
y  sin 1 x
Start with the restricted
y  sin x, 

2
x

2
sine function and reverse
the coordinates of each ordered pair
to produce the graph of the inverse

If 0  x  1, then the value



of sin 1 x is an angle whose


terminal side lies in



sine function.
Quadrant I.
If  1  x  0, then the value 








of sin 1 x is an angle whose
terminal side lies in
Quadrant IV.
Definition
y  sin 1 x
Inverse Sine Function
The inverse sine function, denoted as y  sin 1 x , is the inverse of


y  sin x ,   x  .
2
2
The domain of y  sin 1 x is 1  x  1 and the range is 

2
 y

2
.
(Note that an alternative notation for sin 1 x is arcsin x .)
Do not confuse the notation sin 1 x with  sin x  
1
exponent! Thus, sin 1 x 
1
 csc x . The negative 1 is not an
sin x
1
.
sin x
1
To determine the exact value of sin 1 x , think of the expression sin 1 x as the angle on the interval
  
  2 , 2  whose sine is equal to x. Use conventional angle notation,  , to represent the value of


sin 1 x . If   sin 1 x , then     . Therefore, the terminal side of angle  must lie in
2
2
Quadrant I, in Quadrant IV, on the positive x-axis, on the positive y-axis , or on the negative yaxis.
If   sin
1
x , then 

2
 

2
and the terminal side of angle  lies
in Quadrant I, in Quadrant IV, on
the positive x  axis, on the positive y  axis,
or on the negative y  axis.
Steps for Determining the Exact Value of sin1 x
Step 1. If x is in the interval  1,1 , then the value of sin 1 x must be an angle in the interval   2 , 2  .
Step 2. Let sin 1 x   such that sin   x .
Step 3. If sin   0 , then   0 and the terminal side of angle 
lies on the positive x-axis.
If sin   0 , then 0   

and the terminal side of angle 
2
lies in Quadrant I or on the positive y-axis.
If sin   0 , then 

   0 and the terminal side of angle 
2
lies in Quadrant IV or on the negative y-axis.
Step 4. Use your knowledge of the two special right triangles,
the graphs of the trigonometric functions,
Table 1 from Section 6.4, or Table 2 from Section 6.5
to determine the angle in the correct quadrant whose sine is x.
2
OBJECTIVE 2: Understanding and Finding the Exact and Approximate Values of the Inverse Cosine
Function
Start with the restricted
cosine function and reverse
y  cos x, 0  x  
the coordinates of each ordered pair
to produce the graph of the inverse
y  cos 1 x
cosine function.
If -1  x  0, then the value
of cos1 x is an angle whose
terminal side lies in
Quadrant II.










If 0  x  1, then the value


of cos 1 x is an angle whose


terminal side lies in


Quadrant I.


Definition
Inverse Cosine Function
The inverse cosine function, denoted as y  cos 1 x , is the inverse of
y  cos x , 0  x   .
y  cos 1 x
The domain of y  cos 1 x is 1  x  1 and the range is 0  y   .
(Note that an alternative notation for cos1 x is arccos x .)
3
Think of the expression cos1 x as the angle on the interval  0,   whose cosine is equal to x. It is
important to notice that the interval  0,   is quite different from the interval used to describe the range
of the inverse sine function. It is customary to use the notation,  , to represent the value of cos1 x .
Therefore, if   cos1 x , then 0     . Thus, the terminal side of angle  must lie in Quadrant I,
in Quadrant II, on the positive y-axis, on the positive x-axis, or on the negative x-axis.
If   cos
1
x , then
0  
and the terminal side of angle  lies
in Quadrant I, in Quadrant II, on
the positive y  axis, on the positive
x  axis, or on the negative x  axis.
Steps for Determining the Exact Value of cos1 x
Step 1. If x is in the interval  1,1 , then the value of cos1 x must be an angle in the interval  0,   .
Step 2. Let cos1 x   such that cos  x .

and the terminal side of angle 
2
lies on the positive y-axis.
Step 3. If cos  0 , then  
If cos  0 , then 0   

and the terminal side of angle 
2
lies in Quadrant I or on the positive x-axis.
If cos  0 , then

    and the terminal side of angle 
2
lies in Quadrant II or on the negative x-axis.
Step 4. Use your knowledge of the two special right triangles,
the graphs of the trigonometric functions,
Table 1 from Section 6.4, or Table 2 from Section 6.5
to determine the angle in the correct quadrant whose sine is x.
4
OBJECTIVE 3: Understanding and Finding the Exact and Approximate Values of the Inverse Tangent
Function
y  tan x
y  tan x,  2  x  2
Start with the restricted
tangent function and reverse
the coordinates of each ordered pair
to produce the graph of the inverse
tangent function.
1
y  tan x
If x  0, then the value of tan 1 x
is an angle whose terminal side
lies in Quadrant I.
If x  0, then the value of tan 1 x
is an angle whose terminal side
lies in Quadrant IV.
Definition
Inverse Tangent Function
y  tan 1 x
The inverse tangent function, denoted as y  tan 1 x ,
is the inverse of


y  tan x ,   x  .
2
2
The domain of y  tan 1 x is all real numbers


and the range is   y  .
2
2
(Note that an alternative notation for tan 1 x is arctan x .)
5
  
Think of the expression tan 1 x as the angle on the interval   ,  whose tangent is equal to x. Use
 2 2


conventional angle notation,  , to represent the value of tan 1 x . If   tan 1 x , then     .
2
2
Therefore, the terminal side of angle  must lie in Quadrant I, in Quadrant IV, or on the positive
x-axis.
If   tan
1
x , then 

2
 

2
and the terminal side of angle  lies
in Quadrant I, in Quadrant IV, or
on the positive x  axis.
Steps for Determining the Exact Value of tan1 x
Step 1. The value of tan 1 x must be an angle in the interval   2 , 2  .
Step 2. Let tan 1 x   such that tan   x .
Step 3. If tan   0 , then   0 and the terminal side of angle 
lies on the positive x-axis.
If tan   0 , then 0   

2
and the terminal side of angle 
lies in Quadrant I.
If tan   0 , then 

2
   0 and the terminal side of angle 
lies in Quadrant IV.
Step 4. Use your knowledge of the two special right triangles,
the graphs of the trigonometric functions,
Table 1 from Section 6.4, or Table 2 from Section 6.5
to determine the angle in the correct quadrant whose sine is x.
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7