Notes 2.7
PreCalculus
Section 4.7 Inverse Trig Functions
Name:________________________
Review: Evaluate the following trig Functions:
𝜋
𝜋
1. sin(0)
2. cos( 6 )
3. tan( 4 )
𝜋
4. sec( 3 )
5. cos(
𝜋
2
)
What do you remember about Inverse Functions?
The Inverse of the sine function is called the ___________ function. It can also be written as:
Sine Function:
arcsine function:
For example:
For what 𝜃 does sin(𝜃) = ½
Then, the arcsin( ½) =
Keep in mind that inverse functions can only exist if the original function is one-to-one. Are the arcsine
or cosine functions one-to-one?
Definition of the Inverse Trigonometric Functions
Function
Domain
Range
Definition of the Inverse Trigonometric Functions
Function
Domain
Range
Let’s practice:
√3
2
2.)sin-1 ( )
3.) sin-1 2
4.)arccos 2
√2
5.) cos-1(-1)
6.) arctan 0
7.) tan-1(-1)
8.) arcsin 0
9.) arctan (√3)
1.) arcsin(- ½ )
Composition of functions:
Review: Let f(x)= = x – 1 and g(x) = x2 + 2. How do we find find is f(g(2))?
We can do the same thing to evaluate trigonometric functions:
𝜋
2
Arcsin( cos )
cos(arctan (1))
Inverse functions in the calculator:
Good news! We can use our calculator to determine the inverse trig functions of unfamiliar numbers.
10.) arctan(-8.45)
11.) sin-1 0.2447
12.) arccos 2
For all x in the domain of the function and its inverse…
 x  
f f
1
and f
1
 f x  =
Inverse Properties of Trig Functions
IF __________________________and _____________________________,
Then
sin arcsin x 
and
arcsin sin y  
IF __________________________and _____________________________,
Then
cosarccos x 
and
arccoscos y  
Make sure you
realize that these
properties
are
only valid for the
values that are
acceptable for x
and y!!!
IF __________________________and _____________________________,
Then
tan arctan x 
and
arctan tan y  
If possible, find the exact values.


  

1. tan arctan  
8


2. arctan  tan


3. cos cos 1 (1)

4
How could you evaluate these…


2
3
a. tan  arccos 

 3 
 5 
b. cos arcsin   


HW 2-7
Name:_____________________
Precalculus
Section 4.7 Inverse Functions
Use your notes to fill out the following chart. YOU MUST KNOW THE RANGE FOR EACH INVERSE
FUNCTION FOR THE TEST!!!!
Function
Domain
Range
A. Evaluate the following expressions. Find exact values. i.e NO CALCULATORS!!!



5. cos-1  
 3

 2 


3. arcsin(-1)
 1

 2
7. arcsin(-3.14)
2. arccos 
1. arcos(.5)
3

2 
6. arcsin  



4. sin-1  
3

2 
8. arctan (-1)
B. Evaluate the following composite functions. Use the properties of inverse trigonemtric function if necessary.
Find EXACT VALUES.
  )

 6
9. arcsin(sin  
 3 
)
 4 
13. cos-1(tan 
10. tan(arctan(.8))



14. csc [sin-1  
3
]
2 
11. cos{arccos(-0.1)]
12. arcsin(sin3 )
 1
]
 2
15. tan[cos-1  
C. Use triangles and/or formulas to evaluate the following composite functions
3
]
4
17. sin[arctan 
5
]
 13 
19. cos[arcsin 
18. cos(tan-1 2)
 5
]
 12 
20. tan[arcsin  
  ]

 4
16. cos-1[sin  