IN-CLASS WORK
Sections 7.4 7.5: Inverse Trigonometric Functions
The Sine function is definitely not a one-to-one function (try the horizontal line test) and this means it has no inverse.
y = sin(x)
So first we are going to
“Restrict it’s domain” so we
get a little one-to-one piece of
the function. We will restrict it
to [
]. Note also the
range is [-1,1] 
Now we can define the inverse of this snippet of the sine function. We call this inverse function the Arcsine function.
-1
The function y = arcsin x or y = sin x is the inverse of the function y = sin x with the
restricted domain [
].
Domain of y = arcsin x : [ -1, 1 ]
Range of y = arcsin x : [
]
-1
Remember, the symbol sin x means the inverse function of sin x. It is not the
reciprocal or the multiplicative inverse.
y = arcsin(x)
The purpose of any inverse function is to “undo” what the function does. Since the Sine function takes an angle as an input
and then outputs a number from -1 to 1, the Arcsine function will take numbers from -1 to 1 and output angles between
. It’s a tiny little function and you have to be careful not to use values that are outside its restrictions when you are
working problems. This partial unit circle could help:
So Arcsine is just the “backwards” of Sine (with restrictions). We know
√
√
so
Example 1: Evaluate
(
)
√
( )
We define the inverse of the cosine function similarly, but the one-to-one snippet we use is [0,π]:
y = cos(x)
So first we are going to
“Restrict it’s domain” so we get
a little one-to-one piece of the
function. We will restrict it to
[
]. Note also that the range
is [-1,1] 
(
√
)
IN-CLASS WORK
-1
The function y = arccos x or y = cos x is the inverse of the function y = cos x with the
].
restricted domain [
Domain of y = arccos x : [-1,1]
]
Range of y = arccos x : [
y = arccos(x)
The Arccosine function will take numbers from -1 to 1 and output angles between 0 and π. Like Arcsine, It’s a tiny little
function and you have to be careful not to use values that are outside its restrictions when you are working problems. This
partial unit circle could help:
So Arccosine is just the “backwards” of Cosine (with restrictions).
We know
so
Example 1: Evaluate
(
)
√
( )
Here’s how we define the Inverse Tangent Function:
So first we are going to
“Restrict it’s domain”. We
will grab just the middle
branch so we get a one-toone piece of the function.
We will restrict it to (
).
y = tan(x)
Notice that
and are
not included since tangent
is undefined for those
values anyway.
Now we can define the inverse of this section of the tangent function. We call
this inverse function the Arctangent function.
y = arctan(x)
-1
The function y = arctan x or y = tan x is the inverse of the function y = tan x
with the restricted domain (
).
Domain of y = arctan x : ( -∞, ∞ )
Range of y = arctan x : (
)
Notice the horizontal asymptotes and the
-1
large domain for this function. This is quite
Remember, the symbol tan x means the inverse function of tan x. It is not
different from the arcsine and arccosine
the reciprocal or the multiplicative inverse
functions.
The Arcsine function will take real numbers and output angles between
. Obviously, we aren’t as familiar with the
values of tangent at common angles. This partial unit circle could help:
So Arctangent is just the “backwards” of Tangent (with restrictions). We know
√ so
√
Example 1: Evaluate
(
√
)
( √ )
√
( )