4.7
Inverse Trigonometric
Functions
Definition of the Inverse Trig. Functions
1.
arcsin x
quadrants I and IV
!
!
# " y"
2
2
2.
arccos x
quadrants I and II
0 " y "!
quadrants I and IV
!
!
" < y<
2
2
3.
arctan x
3
Find the values of :
& 1#
arcsin$ ' !
% 2"
I or IV
-1
2
!
" 30° or "
6
sin
!1
3
2
I or IV
!
60° or
3
2
3
1
2
arccos
2
I or II
2
!
45° or
4
1
arccos(! 1)
180° or !
0
1
-1
1
Use your calculator to approximate the value.
Calculator in radians.
arctan( -8.45 )
-1.453001
arcsin( 0.2247 )
.226635
arccos( 2 )
error
Remark: the domain of the inverse functions
of cosine and sine are [-1,1].
!
!
!
Use an inverse trig.
function to write "
as a function of x.
x
cos" =
!5
# x &!
arccos(cos" ) = arccos% (
$5'
# x&
" = arccos% (
$ 5'
x
"
5
Find the exact value.
1
tan[arctan(-5)]
-5
What is the tan of
this triangle? -5
& 5' #
arcsin$ sin
!
3 "
%
5!
3
& & ( ##
!
arcsin$$ sin $ ' ! !! = "
3
% % 3 ""
is coterminal to what
angle that is in the
domain of arcsin?
!
"
3
Find the exact value.
2#
&
tan$ arccos !
3"
%
opp
5
=
=
adj
2
3
5
2
4
&
, 3 )#
cos $arcsin* - '!
+ 5 ("
%
adj 4
=
=
hyp 5
5
-3
Find sin(arccos 3x)
1
0! x!
3
1 = (3x)2 + b2
1
?=
3x
sin(arccos 3x)
cot(arccos 3x) =
2
1! 9x = b
opp
1! 9x
=
=
hyp
1
3x
1! 9x2
2