Download Section 7.4 Inverse Trigonometric Functions I

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 restricting€the domain of y = sin x , −
π
π
2
≤x≤
π
2
, the function is now 1-1 and has an inverse function.
π
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 , −
π
2
≤x≤
π
2
below?
!
€
€
Definition(
Inverse(Sine(Function(
The!inverse(sine(function,!denoted!as! y = sin −1 x ,!is!the!inverse!of!
y = sin x , −
π
2
≤x≤
π
2
.!
!
The!domain!of! y = sin −1 x !is −1 ≤ x ≤ 1 and!the!range!is! −
π
2
≤ y≤
π
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 )
The(negative(1(is(not(an(exponent!((Thus,( sin −1 x ≠
(
1
.(
sin x
−1
=
1
= csc x .(((
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! !$ − π2 , π2 "% .!
Step!2.! Let! sin −1 x = θ !such!that! sin θ = x .!
Step!3.! If! sin θ = 0 ,!then! θ = 0 !and!the!terminal!side!of!angle! θ !lies!on!the!positive!x#axis.!(
If! sin θ > 0 ,!then! 0 < θ ≤
!
π
!and!the!terminal!side!of!angle! θ !
2
!!lies!in!Quadrant!I!or!on!the!positive!y#axis.!
!
!
!
If! sin θ < 0 ,!then! −
!
!
π
2
≤ θ < 0 and!the!terminal!side!of!angle θ !
!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.!!!!!!!!!!!!!!!!!!!!!!!
" 3%
7.4.2 Determine the exact value of the expression sin −1 $ ' .
# 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 ≤ π , the function is now 1-1 and has an inverse function.
Sketch a graph of y = cos x , 0 ≤ x ≤ π , 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 ≤ π , below?
€
€
Definition(
Inverse(Cosine(Function(
The!inverse(cosine(function,!denoted!as! y = cos −1 x ,!!
!!!!!!!!!!!!!!!
is!the!inverse!of y = cos x , 0 ≤ x ≤ π .!!
!
!
The!domain!of! y = cos −1 x !is −1 ≤ x ≤ 1 and!the!range!is!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
0 ≤ y ≤ π .(
(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, π ] .!
Step(2.! Let! cos −1 x = θ !such!that! cos θ = x .!
!
!
!
!
!
!
Step(3.! If!! cos θ = 0 ,!then! θ =
π
2
!and!the!terminal!side!of!!angle θ !lies!on!the!positive!y#axis.!
If! cos θ > 0 ,!then! 0 ≤ θ <
π
!and!the!terminal!side!of!angle! θ !
2
lies!in!Quadrant!I!!or!on!the!positive!x#axis.!!
!
!
If! cos θ < 0 ,!then!
π
2
< θ ≤ π and!the!terminal!side!of!angle! θ !
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!lies!in!Quadrant!II!or!on!the!negative!x#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!cosine!is!x.!!!
" 1 %
7.4.8 Determine the exact value of the expression cos−1 $ −
'.
# 2&
OBJECTIVE 3:
Tangent Function
Understanding and Finding the Exact and Approximate Values of the Inverse
Sketch a graph of y = tan x (draw at least two cycles)
€
•
•
The domain of y = tan x is __________________.
Is the tangent function 1-1? Why? Or why not?______________________
By restricting€the domain of y = tan x , −
function.
€
€
π
2
<x<
π
2
, the function is now 1-1 and has an inverse
π
and
2
π
π
π
π
π
correspond to horizontal asymptotes y = − and y =
of
x = of the graph y = tan x , − < x <
2
2
2
2
2
π
π
€ graph.
the graph of the inverse function to y = tan x , − < x <
. Draw this inverse
2
2
Interchange x’s and y’s from the graph of the principal cycle. The vertical asymptotes x = −
€
!
€
€
€
€
€
Definition(
Inverse(Tangent(Function(
€
The!inverse(tangent(function,!denoted!as! y = tan−1 x,!is!the!inverse!of!
y = tan x , −
π
2
<x<
π
.!
2
€
!
€
y = tan −1 x
The!domain!of!
€
y = tan−1 x!is (−∞,∞)and!
the!range!is! −
π
2
<x<
π
2
.!
€
€
!!!!!!!!!!!!!!(Note!that!an!alternative!notation!for!
tan−1 x !is! arctan x .)!
(€
€
(
€
Steps(for(Determining(the(Exact(Value(of( tan −1 x (
Step(1.! !!!!!The!value!of! tan −1 x !must!be!an!angle!in!the!interval! − π2 , π2 .!
(
)
Step(2.! !!!!!Let! tan −1 x = θ !such!that! tan θ = x .!
Step(3.! !!!!!If! tan θ = 0 ,!then! θ = 0 and!the!terminal!side!of!angle! θ !
!
!
lies!on!the!positive!x#axis.!
!
!!!!!If! tan θ > 0 ,!then! 0 < θ <
!
!!!!!!!!!!!!!!lies!in!Quadrant!I.!
π
2
and!the!terminal!side!of!angle! θ !
!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!
!
π
!
!!!!!If! tan θ < 0 ,!then! −
!
!!!!!!!!!!!!!!!!!lies!in!Quadrant!IV.!!
!
2
< θ < 0 and!the!terminal!side!of!angle! θ !
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!
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!tangent!is!x.!!!
(
(
" 1 %
7.4.13 Determine the exact value of the expression tan −1 $ ' .
# 3&