5.8 Inverse Trigonometric Functions:
Differentiation
recall:
None of the six basic trig. functions
has an inverse function none are one-to-one !
... we restrict their domains
inverse functions
1
notation:
y = sin x
inverse
x = sin y
D: [ ­1 , 1
]
R: [­ 2 , 2 ]
D: [­ 2 , 2 ]
R: [ ­1 , 1 ]
or
y = arcsin x
or
y = sin­1 x
meaning:
"the angle whose sine is x"
2
!
y = sin­1 x
W
KNO
­1 < x < 1
­ < y < 2
2
QI, QIV
­1
y = cos­1
x
­1 < x < 1
0 < y < KNO
W!
QI, QII
y = tan­1 x
W!
O
N
K
­ < x < ­
2 < y < 2
QI, QIV
3
y = cot­1 x
cot­1 x = ­ < x < 0 < y < ­1
­ tan
x
2
1
y = sec­1 x = cos­1 ( )
x
0 < y < ,
y = 2
1
y = csc­1 x = sin­1 ( )
x
­
2 < y < 2 , y = 0 4
Review!
Evaluate in EXACT FORM ­ [No calculator!]
1.
arcsin 1
2.
arctan 3
3.
arccot ­ 3
4.
2 ) tan (arccos 2
5.
5
) cos (arcsin 13
5
Examples:
1.
Write in ALGEBRAIC FORM.
sec (arctan 4x )
SOLVE for x.
1.
arctan ( 2x ­ 3 ) = 4
6
Inverse Trig. Functions: Differentiation
Proof:
[p387 #76 d.)]
d( arccos u) =
dx
­ u'
1­ u2
7
Find the derivative.
Examples:
1.
f(x) = arcsin 4x 2.
g(t) = arccos ( )
t 2 3.
1
2
1
y = x arctan 2x ­ ln
(1
+
4x
) 4
[p387 #56]
8
Assignment
p. 386 # 5 ­25 odd, 29, 31, 33, 37, 41­59 odd, #76a.
9