6.6 – TRIG INVERSES AND
THEIR GRAPHS
Pre-Calc
REVIEW SLIDE
Inverse Trig Functions
Original Function
Inverse
y = sin x
y = sin-1 x
y = arcsin x
y = cos x
y = cos-1 x
y = arccos x
y = tan x
y = tan-1 x
y = arctan x
REVIEW SLIDE
Consider the graph of y = sin x

What is the domain and range of
sin x? Domain: all real numbers
Range: [-1, 1]


What would the graph of
y = arcsin x look like?
What is the domain and range of
arcsin x?
Domain: [-1, 1]
Range: all real numbers
REVIEW SLIDE
Is the inverse of sin x a function?


This will also be true for cosine
and tangent.
Therefore all of the domains are
restricted in order for the inverses
to be functions.
REVIEW SLIDE
How do you know if the domain is
restricted for the original functions?

Capital letters are used to distinguish when the
function’s domain is restricted.
Original Functions with
Restricted Domain
Inverse Function
y = Sin x
y = Sin-1 x
y = Arcsin x
y = Cos x
y = Cos-1 x
y = Arccos x
y = Tan x
y = Tan-1 x
y = Arctan x
REVIEW SLIDE
Original Domains  Restricted Domains
Domain
y = sin x
Range
y = Sin x
y = sin x
y = Sin x
y = cos x
all real numbers
y = Cos x
y = cos x
y = Cos x
y = tan x
all real numbers
except n,
y = Tan x
y = tan x
y = Tan x
all real numbers
all real numbers
all real numbers
where n is an odd
integer
Sketch a graph of y = Sin x
Create a table
x





2

3

4

6
0

6

4

3

2
Remember principal values
y
y = sin(x)
f ( x)

1
3
2
2

2
1

2
0
1
2
2
2
3
2

1






x

Now use your table to generate Sin-1
x





2

3

4

6
0

6

4

3

2
sin 1 ( x )
sin( x )
x
1
1

3
2
2

2
1

2
0
1
2
2
2
3
2

3

2
2

2
1

2
0
1
2
2
2
3
2
1

1



2

3

4

6
0

6

4

3

2
IF YOU CAN REMEMBER AND MEMORIZE WHAT THE original and
inverse funcitons look like, it will make your life a lot easier!!!
Table of Values of Cos x and Arccos x
y = Cos x
X
Y
0
1
π/3
0.5
π/2
0
2π/3
-0.5
π
-1
y = Arccos x
X
Y
1
0
0.5
π/3
0
π/2
-0.5
2π/3
-1
π
The other trig functions require similar restrictions on their domains in order to
generate an inverse.
Like the sine function, the domain of the section of the
tangent that generates the arctan is
Y=Tan(x)
4
  
 2 , 2  .
y
y
Y=Arctan(x)



4


4
4

x
4




4


4
  
D    ,  and R   ,  
 2 2
  
D   ,   and R    , 
 2 2
4
x

Table of Values of Tan x and Arctan x
y = Tan x
X
Y
-π/2
UD
-π/4
-1
0
0
π/4
1
π/2
UD
y = Arctan x
X
Y
UD
-π/2
-1
-π/4
0
0
1
π/4
UD
π/2
Write an equation for the inverse of y = Arctan ½x.
Then graph the function and its inverse.
To write the equation:
1.Exchange x and y
2.Solve for y
x = Arctan ½y
Tan x = ½y
2Tan x = y
Now graph the
original function,
y = Arctan ½x
by switching the
table you just
completed!
Let’s graph 2Tan x = y first.
Complete the table:
Then graph!
1
y = 2Tan x
X
Y
-π/2
Undefined
π/2
-π/2
-π/4
-2
0
0
π/4
2
π/2
Undefined
-1
Now graph the original function, y = Arctan ½x by switching the
table you just completed!
y = 2Tan x
y = Arctan ½x
X
Y
X
Y
-π/2
UD
UD
-π/2
-π/4
-2
-2
-π/4
0
0
0
0
π/4
2
2
π/4
π/2
UD
UD
π/2
π/2
-1
1
-π/2
Write an equation for the inverse of y = Sin(2x).
Then graph the function and its inverse.
y = Sin2x
To write the equation:
1.Exchange x and y
2.Solve for y
x = Sin(2y)
Arcsin(x) = 2y
½Arcsin(x) = y
Let’s graph y = Sin(2x) first.
The domain changes because of the 2,
how?


Divide all
  2x 
2
2
sides by



4
x

2
Y
-π/4
-1
-π/12
-.5
0
0
π/12
.5
π/4
1
4
y = ½Arcsin(x)
1

Now graph the inverse
function, y = Arcsin(x)/2 by
switching the table you just
completed!
X
π/2
-π/2
-1
Remember you can always check and see if
they are symmetric with respect to y = x
X
Y
-1
-π/4
π -.5
0
-π/12
.5
π/12
1
π/4
0



Graph the inverse of:
y
Flip the “x” and “y” and solve for “y”:
x
x
2


 Arc sin y
 Arc sin y
2
  
Sinx   y
 2 
Domain is now:

 
 x 
2
2 2
0 y 
Take the
sine of
both sides
Add π/2 to all
three sides
2
 Arc sin x
  
y  Sinx  
 2 
Let’s find the inverse equation first:



X
Y
0
-1
π/4
 2
π/2
0
3π/4 
π
2
2
2
1

1
π/2
-1
π

Graph
y

2
 Arc sin x
What is the domain for Sin(x)?


 x
2
2
Since we are graphing Arcsin the
y = (π/2)+Arcsin x
domain will become the range,
Now make a
X
Y
but it will change!!

table using the
Solve
for x:
-1
0
y-values as

 2
π/4
your input into
2
y   Arc sin x
2
this function:
0
π/2
y

2
 Arc sin x
  
sin y   x
 2 
Domain is now:

 
 y 
2
2 2
0 y 

Take the
sine of
both sides
2
3π/4
2
1
π

π
y = Arcsinx
π/2
Add π/2 to all
three sides
-1
1
Just shifted up π/2
Now try to graph y  Arc sin x  
4
just by using the shifting technique.

π
y = Arcsinx
π/2
Just shifted down π/4
-1
1
-π/2
Now try to graph y  Arc tan x  
4
just by using the shifting technique.

Just shifted up π/4
π
y = Arctan(x)
π/2
-1
1
-π/2
Graph:

y  Arc tan x 

2
Determine if each of the following is true or false. If
false give a counter example.
1.
Cos-1(cos x) = x for all values of x
x = 270°
FALSE
1.
Sin-1(sin x) = x for all values of x
FALSE
2.
Cos-1(cos 270°) = Cos-1(0) = 90°
Sin-1x + Cos-1x = π/2
x = 180°
Sin-1(sin 180°) = Sin-1(0) = 0°
-1 ≤ x ≤ 1 x = 0
TRUE
3.
Arccos x = Arccos (-x)
FALSE
4.
Tan-1x
= 1/(Tan x)
FALSE
x=0
or try x = 225°
of try x = 1 or -1
Sin-1(0) + Cos-1(0) = 0° + 90°= 90°
-1 ≤ x ≤ 1
x=1
or try x = -1
Arccos(1) ≠ Arccos (-1)
0°
≠ 180°
Tan-1 (0) ≠ 1 / (Tan (0))
0°
≠ 1 / 0 UNDEFINDED
Evaluate each expression
-30 degrees
45 degrees
Evaluate each expression
1
Negative square root
of 3