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Section 7.4 Inverse Trigonometric Functions I
Note: A calculator is helpful on some exercises. Bring one to class for this lecture.
OBJECTIVE 1: Understanding and Finding the Exact and Approximate Values of the Inverse
Sine Function
Sketch a graph of
y = sin x (draw at least two cycles)
y = sin x is^ __________________.
- Is the sine function 1-1? Why? Or why not?______________________
By restricting the domain of
y = sin x ,
π
≤ x ≤
π
, the function is now 1-1 and has an inverse function.
Sketch a graph of
y = sin x ,
π
≤ 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 = sin x ,
π
≤ x ≤
π
below?
!
Definition( Inverse(Sine(Function(
The! inverse(sine(function ,!denoted!as!
1 y sin x
− = ,!is!the!inverse!of!
y = sin x ,
− ≤ x ≤
.!
!
The!domain!of!
1 y sin x
− = !is − 1 ≤ x ≤ 1 and!the!range!is!
− ≤ y ≤
π π
.!
!!!!!!!!!!!!!!(Note!that!an!alternative!notation!for!
1 sin
− x !is arcsin x .)!
CAUTION:((Do(not(confuse(the(notation(
1 sin
−
x (with( ( )
sin csc
sin
x x
x
−
= = .(((
The(negative(1(is(not(an(exponent!((Thus,(
1
sin
sin
x
x
− ≠ .(
(
Steps(for(Determining(the(Exact(Value(of(
− 1 sin x!
Step!1.! If! x" is!in!the!interval! [ −1,1 ],!then!the!value!of!
1 sin
− x must!be!an!angle!in!the!interval! 2 2
π π .!
Step!2.! Let!
1 sin
−
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 !and!the!terminal!side!of!angle! θ!
! !!lies!in!Quadrant!I!or!on!the!positive! y# axis.!
!
!
!
If! sin θ < 0 ,!then! 0
θ 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 .!!!!!!!!!!!!!!!!!!!!!!!
7.4.2 Determine the exact value of the expression sin
− 1
OBJECTIVE 2: Understanding and Finding the Exact and Approximate Values of the Inverse
Cosine Function
Sketch a graph of
y = cos x (draw at least two cycles)
y = cos x is __________________.
- Is the cosine function 1-1? Why? Or why not?______________________
!
Step(3.! If!! cos θ = 0 ,!then!
θ !and!the!terminal!side!of!!angle θ !lies!on!the!positive! y# axis.!
If! cos θ > 0 ,!then! (^0)
θ !and!the!terminal!side!of!angle! θ!
lies!in!Quadrant!I!!or!on!the!positive! x# axis.!!
!
!
If! cos θ < 0 ,!then!
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 .!!!
7.4.8 Determine the exact value of the expression cos
− 1 −
.
OBJECTIVE 3: Understanding and Finding the Exact and Approximate Values of the Inverse
Tangent Function
Sketch a graph of
y = tan x (draw at least two cycles)
y = tan x is __________________.
- Is the tangent function 1-1? Why? Or why not?______________________
By restricting the domain of
y = tan x ,
π
< x <
π
, the function is now 1-1 and has an inverse
function.
1
tan
−
y = x
Interchange x ’s and y ’s from the graph of the principal cycle. The vertical asymptotes
x = −
and
x =
of the graph
y = tan x ,
π
< x <
π
correspond to horizontal asymptotes
y = −
and
y =
of
the graph of the inverse function to
y = tan x ,
π
< x <
π
. Draw this inverse graph.
!
Definition( Inverse(Tangent(Function(
The! inverse(tangent(function ,!denoted!as!
y = tan
− 1 x ,!is!the!inverse!of!
y = tan x ,
π
< x <
π
!
The!domain!of!
y = tan
− 1 x !is
(^ −∞ ,∞)and!
the!range!is!
π
< x <
π
.!
!!!!!!!!!!!!!!(Note!that!an!alternative!notation!for!
tan
− 1 x !is!
arctan x .)!
(
(
Steps(for(Determining(the(Exact(Value(of(
1 tan x
− (
Step(1.! !!!!!The!value!of!
1 tan
−
x !must!be!an!angle!in!the!interval! ( )
2 2
π π .!
Step(2.! !!!!!Let!
1 tan
−
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 and!the!terminal!side!of!angle!^ θ^!^ !!!!!!!!!!!!!!!!!!!!!!!!!!!!
! !!!!!!!!!!!!!!lies!in!Quadrant!I.!
!
!
! !!!!!If! tan θ < 0 ,!then! 0
θ and!the!terminal!side!of!angle! θ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! !!!!!!!!!!!!!!!!!lies!in!Quadrant!IV.!!
!
