4.7 INVERSE TRIGONOMETRIC
FUNCTIONS
For an inverse to exist the function MUST
be one- to - one
• A function is one-to• So
one if for every x there • If x and/or y is raised
is exactly one y and
to an even power then
for every y there is
the inverse does not
exactly one x.
exist unless the
domain is restricted.
• The equation y = x2
• In order to restrict the
domain, a basic
• does not have an
inverse because two
knowledge of the
different x values will
shape of the graph is
produce the same ycrucial. This is a
value.
parabola with (0,0) as
• i.e. x = 2 and x = -2 will
the vertex. Restrict
produce y = 4.
the domain to the
• The horizontal line
interval [0,infinity) to
test fails.
make it one-to-one.
Now let’s look at the trig functions
y
y = cos x
y
y = sin x


x
x
















y







y = tan x
















x






y

x








Not a 1-1 function
So it currently does
not have an inverse

For the graph of y = sin x, the Domain is (-∞, ∞)
the Range is [-1, 1]
Now it’s 1-1!
y

x









However we can restrict the domain to [-/ , /]
Note the range will remain [-1, 1]
y = sinx
y

x




The inverse of sinx
or
sin1 x
Is denoted as arcsinx



On the unit circle:
y

x

For the inverse sine function with
angles only from -/ to /
our answers will only be in either
quadrant 1 for positive values and
quadrant 4 for negative values.

Find the exact value, if possible,

 1
arcsin   
 2
sin 1
3
2
sin -1 2
y

x









y = cos x is not one to one, so its domain will also need to be
restricted.
y = cos x is not one to one, so its domain will also need to be
restricted.
y
y = cos x

On this interval, [0, ] the
cosine function is one-toone and we can now
define the inverse cosine
function.
y = arccos x or y = cos-1 x
x
/
/
/

/

/
/

y = arccos x
On the unit circle ,
inverse cosine will only
exist in quadrant 1 if the
value is positive and
quadrant 2 if the value is
negative.
y

x

Find the exact value for:

arccos

2
2
arccos(1)
 3
cos -1  

2


y = tan x
y = tanx
y

Remember that tangent is undefined at
-/ and /








/
/
/

x
/
/
/










y = arctanx
y

Remember that tangent is undefined at
-/ and /








/
/
/

x
/
/
/










arctan  1
Find the exact value
tan 1 0

3
arctan  

3


Using the calculator.
•
•
•
•
•
•
Be in radian mode
Arctan(-15.7896)
Arcsin(.3456)
Arccos(-.6897)
Arcsin(1.4535)
Arccos(-2.4534)
H Dub
• 4-7 Page 349 #1-16all, 49-67odd