Pre-Calculus Honors Section 4.7 Inverse Trigonometric Functions
Objective: To evaluate the six trigonometric functions.
Lesson:
Recall from Section 1.5 that for a function to have an inverse it must pass the
horizontal line test.
Let’s consider the sine function.
It does not pass the horizontal
line test.
However if you restrict the


domain from  2  x  2 the following apply.
 


1. From the interval  ,  the function is increasing.
 2 2
 


2. From the interval  ,  sin x ranges from  1  sin x  1 (or sin x can
 2 2
be anywhere between -1 and 1)
 


3. From the interval  ,  sin x passes the horizontal line test.
 2 2
 


Therefore in the interval  ,  sin x has an inverse that can be denoted in the
 2 2
following way:
y  sin 1 x or y  arcsin x
1
Remember: sin x means inverse not reciprocal
Definition of Inverse Sine Function
y  arcsin x if and only if sin y  x


Where 1  x  1 and  2  y  2
Domain: [-1, 1]
  
Range:  2 , 2 


1
sin x
y  sin x
y  sin 1 x
Definition of Inverse Cosine Function
y  arccos x if and only if cos y  x
y  cos x
y  cos 1 x
y  tan x
y  tan 1 x
Where 1  x  1 and 0  y  
Domain: [-1, 1]
Range: 0,  
Definition of Inverse Cosine Function
y  arctan x if and only if tan y  x
Domain: (, )
  
Range:  2 , 2 


Ex. 1) Evaluate the expression.
 1
arcsin
  
a)
 2
1
2
Think about the problem as if its asking, what is the angle whose sine is  ?
1
2
In other words sin x   , what is x?
Remember there can only be one correct answer and it has to be from Quadrant I
(positive values) or Quadrant IV (negative values) for the sine function.
 3


b) sin  2  


1
c) sin 2 
 2


arccos
d)
 2 


e) arccos1 
1
Remember there can only be one correct answer and it has to be from Quadrant I
(positive values) or Quadrant II (negative values) for the cosine function.
g) arctan 1 
f) arctan 0 
Remember there can only be one correct answer and it has to be from Quadrant I
(positive values) or Quadrant IV (negative values) for the tangent function.
Ex. 2) Write each trigonometric function in inverse function form or vice versa.

sin
1
a)
2
 3 


arccos
c)
 2  6


3
 
sin



b)
3 2
Assignment page 327 #1-9 odds and #14
No not use a calculator for this assignment
Pre-Calculus Honors Section 4.7 Inverse Trigonometric Functions
Day 2
Objectives:
1. To evaluate inverse trig. Functions using a calculator.
2. To evaluate compositions of trig. Functions.
Lesson:
Composition of Trigonometric Functions
1
1
Recall: f ( f ( x))  x and f ( f ( x))  x , likewise
 sin(arcsin x)  x
and

and
arccos(cos y )  y
If 0  y  
and
arctan(tan y )  y
If 1  x  1
 tan(arctan x)  x
If    x  

If  2  y  2
If 1  x  1
 cos(arccos x)  x
arcsin(sin y )  y


If  2  y  2
Ex. 1) Use a calculator to approximate the value of the expression to the nearest
hundredth. (Remember to be in radian mode unless there is a degree sign.)
a) arcsin 0.45
b) arctan 18
Ex. 2) Use an inverse trigonometric function to write  as a function of x.
a)
5
x2

Ex. 3) Use the properties of inverse functions to evaluate the expression.
a) tanarctan  5 
(Remember tan and arctan are inverses of each other)
 5 
arcsin
 sin

b)
3


 


For arcsin your answers have to be between  ,  .
 2 2
1
c) cos(cos  ) 
For cos your answers have to be between [1, 1] .
Ex. 4) Find the exact value of the expression. Hint: make a sketch of a right
triangle.
4

sin
arctan


a)
3



 3 
sec
arctan

    

b)
 5 

If the arctan is negative in what Quadrant?

 3 
cos
arcsin

    

c)
 5 

Assignment page 328 #15-61 odds