Download Section 4.7

Topic/Objective:
Name:
4-7 INVERSE TRIGONOMETRIC
FUNCTIONS
Class/Period:
Date:
Essential Question: How do you use the inverse trigonometric functions to solve problems?
Questions:
INVERSE FUNCTIONS. A function must be a one-toone function to have an inverse.
HORIZONTAL LINE TEST. If a horizontal line is
drawn through the graph of a function and the
horizontal line only intersects the graph of the function
once, then the function is one-to-one and has an
inverse.
SINE FUNCTION. The graph shows three periods of
the basic sine function y  sin x . It is obvious that the
sine function fails the horizontal line test.
RESTRICTED SINE FUNCTION. If the domain of
the basic sine function is restricted to the interval
x    ,   , the function has the following
 2 2
properties.
1. The function is increasing on the interval
  ,  .
 2 2
2. The range of the function is  1,1 .
3. On the interval x    ,   , the basic sine
 2 2
function has a unique inverse called the inverse sine
function.
INVERSE SINE FUNCTION. The inverse sine
function, often called the arcsine function, is denoted
as y  sin 1 x or y  arcsin x . The domain of
y  arcsin x is  1,1 and the range is    ,   .

2
2
EXAMPLE 1. Use the Unit Circle to evaluate each inverse sine function, if possible.
 3
 1
a. arcsin   
b. sin 1 

 2
 2 
1
Questions: c. sin 1  2
INVERSE COSINE FUNCTION. The
inverse cosine function is found by
restricting the domain of the cosine
function to x  0,   and is denoted as
y  cos1 x or y  arccos x .
DOMAIN/RANGE OF THE INVERSE
COSINE FUNCTION
Domain:  1,1
Range:  0,  
INVERSE TANGENT FUNCTION.
The inverse tangent function is found by
restricting the domain of the tangent
function to x    ,   and is
 2 2
denoted as y  tan 1 x or y  arctan x .
DOMAIN/RANGE OF THE INVERSE
TANGENT FUNCTION
Domain:  1,1
Range:    ,  
 2 2
EXAMPLE 2. Evaluate each inverse trigonometric function using the calculator.
 2
a. arccos 
b. arccos  1

 2 
c. arctan  0
d. arctan  1
e. tan 1  8.45
f. cos1  2 
COMPOSITION OF FUNCTIONS. The composition of functions is denoted by
 f g  x  or f  g  x   . If f  f 1  x    x or f 1  f  x    x , then the functions are
inverses of each other. This property leads to the properties of inverse trigonometric
functions.
INVERSE PROPERTIES
1. If 1  x  1 and    y   , then sin  arcsin  x    x and arcsin  sin  y    y .
2
2
2. If 1  x  1 and 0  y   , then cos  arccos  x    x and arccos  cos  y    y
2
Questions:
3. If x is a real number and  
arctan  tan  y    y .
2
 y 
2
, then tan  arctan  x    x and
EXAMPLE 3. Use the calculator to evaluate each function, if possible.
  5  
a. tan  arctan  5  
b. arcsin  sin   
  3 
c. cos  cos 1   
EVALUATE COMPOSITIONS OF TRIGONOMETRIC FUNCTIONS
ALGEBRAICALLY. To evaluate composition of trigonometric functions
algebraically, use the following procedures.
1. Identify the innermost trigonometric function and let this function equal u .
2. Identify the Quadrant in which the terminal side of u lies.
3. Sketch a triangle, in standard position, modeling the problem.
4. Use the Pythagorean Theorem to find the length of the each side of the triangle.
5. Use the triangle to evaluate the outermost trigonometric function.
EXAMPLE 4. Evaluate the following trigonometric function compositions.


 2 
 3 
a. tan  arccos   
b. cos  sin 1    
 3 
 5 


EVALUATE COMPOSITIONS OF FUNCTIONS: A CALCULUS APPROACH.
Using techniques from Calculus, trigonometric compositions reduce to algebraic
functions that do not involve trigonometry at all.
EXAMPLE 5. Given the triangle shown in the diagram, write the composition
function tan  and cot  as algebraic expressions.
3
Questions:
EXAMPLE 6. Write the following trigonometric compositions as algebraic
expressions.
1
a. sin  arccos  3 x   , 0  x 
3
b. cot  arccos  3x   , 0  x 
1
3
SUMMARY:
4