Download Section 7.5 Inverse Trigonometric Functions II

Section 7.5 Inverse Trigonometric Functions II
Note: A calculator is helpful on some exercises. Bring one to class for this lecture.
OBJECTIVE 1:
Evaluating composite Functions involving Inverse Trigonometric Funcitons of
−1
the Form f ! f and f −1 ! f
It is imperative that you know and understand the three inverse trigonometric functions introduced in 7.4.
€ y = sin−1 x€ (Say: “y is the angle whose sine is x”)
A.
1. Draw the graph of the inverse sine function.
€
2. Domain___________________
Range _________________
3. The range of the inverse sine function represents an angle whose terminal side lies in
a. ____________________ (which quadrants)
b. ___________________________________ (which axes)
c. Show this with a graph.
B. y = cos−1 x
(Say: “y is the angle whose cosine is x”)
1. Draw the graph of the inverse cosine function.
€
2. Domain___________________
Range _________________
3. The range of the inverse cosine function represents an angle whose terminal side lies in
a. ____________________ (which quadrants)
b. ___________________________________ (which axes)
c. Show this with a graph.
C. y = tan−1 x
(Say: “y is the angle whose tangent is x”)
1. Draw the graph of the inverse tangent function.
€
2. Domain___________________
Range _________________
3. The range of the inverse tangent function represents an angle whose terminal side lies in
a. ____________________ (which quadrants)
b. ___________________________________ (which axes)
c. Show this with a graph.
CAUTION: For trigonometric expressions of the form ( f ! f −1 )(x) or ( f −1 ! f )(x) , the cancellation
equations work ONLY if x is in the domain of the “inner” function.
!
€
€
Cancellation*Equations*for*Compositions*of*Inverse*Trigonometric*Functions*
Cancellation*Equations*for*the*Restricted*Sine*Function*and*its*Inverse!
sin sin −1 x = x !for!all!x"in!the!interval! [ −1,1] !
(
)
sin −1 ( sin θ ) = θ !for!all! θ !in!the!interval! !$ − π2 , π2 "% .!!*
Cancellation*Equations*for*the*Restricted*Cosine*Function*and*its*Inverse!
cos cos −1 x = x !for!all!x"in!the!interval! [ −1,1] !
(
)
cos −1 ( cos θ ) = θ !for!all! θ "in!the!interval! [0, π ] .*
Cancellation*Equations*for*the*Restricted*Tangent*Function*and*its*Inverse!
tan tan −1 x = x !for!all!x"in!the!interval! ( −∞, ∞ ) .!
(
)
tan −1 ( tan θ ) = θ !for!all! θ "in!the!interval! ( − π2 , π2 ) .*
EXAMPLES: Find the exact value of each expression or state that it does not exist.
"
" 3 %%
7.5.4 sin $$ sin −1 $ − '''
# 2 &&
#
"
" 8 %%
7.5.8 cos $ cos−1 $ − ''
# 5 &&
#
" π%
7.5.9 sin −1 $ sin '
# 7&
"
3π %
7.5.13 cos−1 $ cos '
#
4 &
" 7π %
7.5.14 tan −1 $ tan
'
#
6 &
" 8π %
7.5.17 tan −1 $ tan '
#
3 &
" 13π %
7.5.26 cos−1 $ cos
'
#
10 &
OBJECTIVE 2:
Evaluating composite Functions involving Inverse Trigonometric Funcitons of
−1
the Form f ! g and f −1 ! g
Method:
€ 1. Evaluate
€ the “inner expression” and then evaluate the “outer expression.”
2. It may be necessary to draw a triangle (using x, y, or r) in the appropriate quadrant (depending
on if the trig value is positive or negative), determine the value of the missing side and write the
trig function requested in the “outer expression.”
3. If an exact value of the “inner expressions” cannot be determined, try writing the expressions as
an equivalent expression using a cofunction identity.
EXAMPLES. : Find the exact value of each expression or state that it does not exist.
7.5.32 cos ( tan −1 (−1))
"
" −1 %%
7.5.34 tan $ sin −1 $
''
# 2 &&
#
"
" 3 %%
7.5.37 tan $$ sin −1 $ '''
# 4 &&
#
" " 5π %%
7.5.43 cos−1 $ sin $ ''
# # 4 &&
" " 19π %%
7.5.46 cos−1 $ sin $
''
# # 9 &&
OBJECTIVE 3:
Functions
Definition*
Understanding the Inverse cosecant, Inverse Secant, and Inverse Cotangent
Inverse*Cosecant*Function*
The!inverse*cosecant*function,!denoted!as! y = csc −1 x ,!is!the!inverse!
!of! y = csc x, !$ − π2 , 0 U 0, π2 "% .!!
) (
The!domain!of! y = csc −1 x !is ( −∞, −1] U [1, ∞ ) !
y = csc −1 x
and!the!range!is! !$ − π2 , 0 U 0, π2 "% .!
) (
*
Definition*
Inverse*Secant*Function*
The!inverse*secant*function,!denoted!as! y = sec −1 x ,!is!the!inverse!of
y = sec −1 x
y = sec −1 x
y = sec x, !#0, π2 ) U ( π2 , π "$ .!!!*
The!domain!of! y = sec −1 x !is ( −∞, −1] U [1, ∞ ) and!the!range!is!
!#0, π2 ) U ( π2 , π "$ .!
!
!
!
Definition*
y = cot −1 x
Inverse*Cotangent*Function*
*
The!inverse*cotangent*function,!denoted!as!!
y = cot −1 x ,!!is!the!inverse!of!
y = cot x, 0 < x < π .!!!
The!domain!of! y = cot −1 x !is! ( −∞, ∞ ) and!the!range!is! − π2 , 0 U 0, π2 "$ .!
(
) (
!
!
EXAMPLES. : Find the exact value of each expression or state that it does not exist.
7.5.48 csc−1 (2)
" 1 %
7.5.49 cot −1 $ − '
# 3&
OBJECTIVE 4:
Writing Trigonometric Expressions as Algebraic Expressions Functions
In Calculus, it is often necessary to write trigonometric expressions algebraically. In this text u is used as
the unknown variable. In calculus x is often (but not always) used.
We assume that the variable represents an angle whose terminal side is located in Quadrant I
Method:
€
1. Given the inverse trigonometric expression (inner expression which represents an unknown angle
θ ), draw the triangle represented with θ in standard position and the terminal side located in QI
2. Label the given sides of the triangle. Since trigonometric expressions represent ratios of the sides
of right triangles, two sides are always given.
3. Determine algebraically the 3€rd side.
4. Write the expression asked for (outside expression).
EXAMPLES. Rewrite each trigonometric expression as an algebraic expression involving the variable u.
Assume that u > 0 and that the value of the “inner” trigonometric expression represents an angle θ such
π
that 0 < θ < .
2
€
€
7.5.54 tan(cos−1 2u)
€
"
3%
7.5.55 cos $ sin −1 '
#
u&
"
%
u
7.5.57 sec $ sin −1
'
#
u 2 +121 &