5.7
Inverse Trigonometric
Functions
Definition of the Inverse Trig. Functions
1.
arcsin x
quadrants I and IV
2.
arccos x
quadrants I and II
3.
arctan x
quadrants I and IV
p
p
- £ y£
2
2
0 £ y £p
p
p
- < y<
2
2
3
Find the values of :
Ê 1ˆ
arcsinÁ - ˜
Ë 2¯
sin
-1
3
2
I or IV
-1
2
p
- 30! or 6
I or IV
p
60! or
3
2
3
1
2
arccos
2
I or II
p
45! or
4
2
1
arccos(- 1)
180! or p
0
1
-1
1
Find the exact value.
2ˆ
Ê
tanÁ arccos ˜
3¯
Ë
opp
5
=
=
adj
2
3
5
2
È
Ê 3 ˆ˘
cos ÍarcsinÁ - ˜˙
Ë 5 ¯˚
Î
adj 4
=
=
hyp 5
4
5
-3
Find sin(arccos 3x)
1
0£ x£
3
1 = (3x)2 + b2
1
?=
3x
sin(arccos 3x)
cot(arccos 3x) =
1- 9x = b
2
opp
1- 9x
=
=
hyp
1
3x
1- 9x2
2
Derivatives of the Inverse Trigonometric Functions
pg. 375
3
3
=
2
2
1+ 9x
1 + (3 x)
-1 2
d
1
1
2
x
(
)
arcsin x =
=
dx
2 x 1- x
1- x
d
[arctan(3x)]=
dx
[
]
2x
2e
d
2x
=
arc
sece
=
[
]
2x
2x 2
dx
e (e ) -1
2
e -1
4x
Differentiate
y = arcsin x + x 1 - x
2
Ê1ˆ
2 -1 2
2
y'=
+
x
-2x
1x
+
1x
Á
˜
(
)
(
)
2
Ë2¯
1- x
1
1
=
1- x
=
(
2
-
2 1- x
2
1- x
2
x
2
1- x
) =2
+ 1- x =
2
2
1- x
2
1- x +1- x
2
1- x
2
2