Section 7.4 Inverse Trigonometric Functions I
[Pages:6]Section 7.4 Inverse Trigonometric Functions I
Note: A calculator is helpful on some exercises. Bring one to class for this lecture.
OBJECTIVE 1: Sine Function
Understanding and Finding the Exact and Approximate Values of the Inverse
Sketch a graph of y = sin x (draw at least two cycles)
? The domain of y = sin x is __________________. ? Is the sine function 1-1? Why? Or why not?______________________
By
restrictingthe
domain
of
y
= sin x ,
- 2
x
2
, the function is now 1-1 and has an inverse function.
Sketch a graph of y = sin x , - x , plotting end points and several other points.
2
2
Interchange x's and y's from the graph above. Are the points on the graph of the inverse function to
y = sin x , - x below?
2
2
!
Definition( Inverse(Sine(Function(
The!inverse(sine(function,!denoted!as! y = sin-1 x ,!is!the!inverse!of!
y
=
sin
x
,
-
2
x
2
.!
!
The!domain!of!
y
=
sin -1
x
!is
-1
x
1
and!the!range!is!
-
2
y
2
.!
!!!!!!!!!!!!!!(Note!that!an!alternative!notation!for! sin-1 x !is arcsin x .)!
CAUTION:((Do(not(confuse(the(notation( sin-1
x
(with( (sin
x )-1
=
1 sin
x
=
csc
x
.(((
The(negative(1(is(not(an(exponent!((Thus,(
sin-1
x
1 sin
x
.(
(
Steps(for(Determining(the(Exact(Value(of( sin-1 x !
[ ] Step!1.! If!x"is!in!the!interval!
-1,1
,!then!the!value!of!
sin
-1
x
must!be!an!angle!in!the!interval!
!$
-
2
,
2
"%
.!
Step!2.! Let! sin-1 x = !such!that! sin = x .!
Step!3.! If! sin = 0 ,!then! = 0 !and!the!terminal!side!of!angle! !lies!on!the!positive!x#axis.!(
If! sin
>
0
,!then!
0
<
2
!and!the!terminal!side!of!angle!
!
!
!!lies!in!Quadrant!I!or!on!the!positive!y#axis.!
!
!
!
If! sin
<
0
,!then!
-
2
<
0
and!the!terminal!side!of!angle
!
!
!lies!in!Quadrant!IV!or!on!the!negative!y#axis.!!
!
!
!
!
Step!4.! Use!your!knowledge!of!the!two!special!right!triangles!and!the!graphs!of!the!trigonometric!functions,!!to!!! !!!!!!!!!!!!!!!determine!the!angle!in!the!correct!quadrant!whose!sine!is!x.!!!!!!!!!!!!!!!!!!!!!!!
" 7.4.2 Determine the exact value of the expression sin-1 $
3
% '
.
#2&
OBJECTIVE 2: Cosine Function
Understanding and Finding the Exact and Approximate Values of the Inverse
Sketch a graph of y = cos x (draw at least two cycles)
? The domain of y = cos x is __________________. ? Is the cosine function 1-1? Why? Or why not?______________________
By restricting the domain of y = cos x , 0 x , the function is now 1-1 and has an inverse function.
Sketch a graph of y = cos x , 0 x , plotting end points and several other points.
Interchange x's and y's from the graph above. Are the points on the graph of the inverse function to y = cos x , 0 x , below?
Definition( Inverse(Cosine(Function(
The!inverse(cosine(function,!denoted!as! y = cos-1 x ,!! is!the!inverse!of y = cos x , 0 x .!!
!
The!domain!of! y = cos-1 x !is -1 x 1and!the!range!is!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 0 y .(
(Note!that!an!alternative!notation!for! cos-1 x !is arccos x .)!
! !
!
!!!!!!!!!!!!!!! !
Steps(for(Determining(the(Exact(Value(of( cos-1 x (
Step(1.! If!x"is!in!the!interval! [-1,1] ,!then!the!value!of! cos-1 x must!be!an!angle!in!the!interval! [0, ].!
Step(2.! Let! cos-1 x = !such!that! cos = x .!
! ! ! ! !
!
Step(3.!
If!! cos
=
0
,!then!
=
2
!and!the!terminal!side!of!!angle
!lies!on!the!positive!y#axis.!
If! cos
>
0
,!then! 0
<
2
!and!the!terminal!side!of!angle!
!
lies!in!Quadrant!I!!or!on!the!positive!x#axis.!!
!
!
If! cos
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