Transcript Inverse Trigonometric Functions

Sec. 4.7a
Reminders…
What does it mean for a function to be one-to-one???
 It passes both the vertical and horizontal line tests
What does it mean for a function to pass the HLT???
 It has an inverse that is also a function
Are any of the six basic trigonometric functions one-to-one???
NOPE!!!
 If not, then how do they have inverse functions?
We first restrict the domain of the basic function…
Inverse Sine Function=arcsine
We restrict the domain of
1
y  sin x to the interval    2,  2

2


2
–1

2
–1

1

2
The unique angle y in the interval    2,  2 such that
sin y  x
sin 1 x
is the inverse sine (or arcsine) of x, denoted
arcsin x .
1
The domain of y  sin x
or
is
1,1 and the range is

2,  2
Guided Practice
Find the exact value of each expression without a calculator.
1  1 
1. sin  
2
1
Note: The values of y  sin x will always
be found on the right-hand side of the unit
circle, between   2 .
We find the point on the right half of the unit circle whose
y-coordinate is 1/2 and draw a reference triangle:
1
2
Do we recognize this angle???
1 
 sin   
2 6
1
Guided Practice
Find the exact value of each expression without a calculator.


3
2. sin  
 2    3


1
Draw another reference triangle, and identify the angle…
 
3. sin   DNE
2
1

 1 , so…
The domain of the inverse sine function is [–1,1], and
2
Guided Practice
Find the exact value of each expression without a calculator.
    
4. sin  sin    
  9  9
1

sin
9
Draw an angle of  9 in standard
position and mark its y-coordinate on
the y-axis. The angle in the interval
  2,  2 whose sine is this

number is 

9.
Guided Practice
Find the exact value of each expression without a calculator.
  5
5. sin  sin 
  6
1
5
6
 
 
 6
Draw an angle of 5 6 in standard
position and mark its y-coordinate on
the y-axis. The angle in the interval
  2,  2 whose sine is this

number is

  5 6   6 .
Inverse Cosine Function=arccosine
We restrict the domain of y
 cos x to the interval  0,  

1

–1
–1

1

The unique angle y in the interval 0,  such that
cos y  x is the inverse cosine (or arccosine) of x, denoted
cos 1 x
or
arccos x .
The domain of
1
y  cos x
is
1,1 and the range is 0,  
Inverse Tangent Function=arctangent
We restrict the domain of y
 tan x
to the interval
 
2,  2

2


2

2


2
The unique angle y in the interval    2,  2  such that
tan y  x is the inverse tangent (or arctangent) of x, denoted
tan 1 x
or
arctan x .
The domain of
1
y  tan x
is
 ,   and the range is
 
2,  2
Guided Practice
Find the exact value of each expression without a calculator.
1


y

cos
x will always
2 Note: The values of
1
6. cos  
 2  be found on the top half of the unit circle,

 between 0 and .
We find the point on the top half of the unit circle whose
x-coordinate is  2 2 and draw a reference triangle:
Do we recognize this angle???
2

2

2  3
cos  
 
 2  4
1
Guided Practice
Find the exact value of each expression without a calculator.
7.
tan
1
1
Note: The values of y  tan x will always
be found on the right-hand side of the unit
circle, between (but not including)   2.
3
We find the point on the right side of the unit circle whose
y-coordinate is 3 times its x-coordinate and draw a
reference triangle:
3
2
1
2
Do we recognize this angle???
tan
1
3

3
Guided Practice
Find the exact value of each expression without a calculator.
8. cos1
cos  1.1  1.1
Draw an angle of –1.1 in standard position, and mark its
x-coordinate on the x-axis. The angle in the interval 0, 
whose cosine is this number is 1.1…

cos  1.1
