Download Section 7.4 Inverse Trigonometric Functions I

Section 7.4 Inverse Trigonometric Functions I
Note: A calculator is helpful on some exercises. Bring one to class for this lecture.
OBJECTIVE 1:
Sine Function
Understanding and Finding the Exact and Approximate Values of the Inverse
Sketch a graph of y  sin x (draw at least two cycles)



The domain of y  sin x is __________________.
Is the sine function 1-1? Why? Or why not?______________________
By restrictingthe domain of y  sin x , 


2
x

2
, the function is now 1-1 and has an inverse function.

Sketch a graph of y  sin x ,   x  , plotting end points and several other points.
2
2




Interchange x’s and y’s from the graph above. Are the points on the graph of the inverse function to
y  sin x , 



2
x

2
Definition
below?
Inverse Sine Function
The inverse sine function, denoted as y  sin1 x , is the inverse of
y  sin x , 

2
x

2
.
The domain of y  sin1 x is 1 x 1and the range is 

2
 y

2
.
(Note that an alternative notation for sin1 x is arcsin x .)
CAUTION: Do not confuse the notation sin1 x with sin x 
1
The negative 1 is not an exponent! Thus, sin1 x 
1
.
sin x

1
 csc x .
sin x
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   

2
and the terminal side of angle 
lies in Quadrant I or on the positive y-axis.
If sin   0, then 

2
   0 and the terminal side of angle 
lies in Quadrant IV or on the negative y-axis.
Step 4. Use your knowledge of the two special right triangles and the graphs of the trigonometric functions, to
determine the angle in the correct quadrant whose sine is x.
7.4.2 Determine the exact value of the expression_____________
7.4.4 Use a calculator to approximate the value of the expression ______________
OBJECTIVE 2:
Cosine Function
Understanding and Finding the Exact and Approximate Values of the Inverse
Sketch a graph of y  cos x (draw at least two cycles)



The domain of y  cos x is __________________.
Is the cosine function 1-1? Why? Or why not?______________________

By restricting the domain of y  cos x , 0  x   , the function is now 1-1 and has an inverse function.
Sketch a graph of y  cos x , 0  x   , plotting end points and several other points.




Interchange x’s and y’s from the graph above. Are the points on the graph of the inverse function to
y  cos x , 0  x   , below?


Definition
Inverse Cosine Function
The inverse cosine function, denoted as y  cos1 x,
is the inverse of y  cos x , 0 x   .
The domain of y  cos1 x is 1 x 1and the range is
0 y  .
(Note that an alternative notation for cos1 x is arccosx.)
1
Steps for Determining the Exact Value of cos 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.
Step 3. If cos  0, then  

and the terminal side of angle  lies on the positive y-axis.
2
If cos  0, then 0   

2
and the terminal side of angle 
lies in Quadrant I or on the positive x-axis.
If cos  0, then

2
    and the terminal side of angle 
lies in Quadrant II or on the negative x-axis.
Step 4. Use your knowledge of the two special right triangles and the graphs of the trigonometric functions to
determine the angle in the correct quadrant whose cosine is x.
7.4.8 Determine the exact value of the expression____________
7.4.12 Use a calculator to approximate the value of the expression__________________
OBJECTIVE 3:
Tangent Function
Understanding and Finding the Exact and Approximate Values of the Inverse
Sketch a graph of y  tan x (draw at least two cycles)



The domain of y  tan x is __________________.
Is the tangent function 1-1? Why? Or why not?______________________
By restrictingthe domain of y  tan x , 
function.



2
x

2
, the function is now 1-1 and has an inverse
Interchange x’s and y’s from the graph of the principal cycle. The vertical asymptotes x  
x

2
of the graph y  tan x , 

2
x

2
correspond to horizontal asymptotes y  
the graph of the inverse function to y  tan x , 



2
x

2

2

2
and
and y 

 graph.
. Draw this inverse




Inverse Tangent
Function
Definition
The inverse tangent function, denoted as y  tan1 x , is the inverse of
y  tan x , 


2
x

2
.

y  tan1 x
 domain of y  tan1 x is ,and
The
the range is 


2
x

2
.

(Note that an alternative notation for tan1 x is arctan x .)



Steps for Determining the Exact Value of tan1 x


Step 1.
1
The value of tan x must be an angle in the interval  2 , 2 .
Step 2.
1
Let tan 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.
2
of
Step 4.
Use your knowledge of the two special right triangles and the graphs of the trigonometric
functions to determine the angle in the correct quadrant whose tangent is x.
7.4.13 Determine the exact value of the expression_____________
7.4.17 Use a calculator to approximate the value of the expression ______________