Download Section 7.5 - Gordon State College

Section 7.6
Inverse Trigonometric
Functions
THE INVERSE SINE FUNCTION
Definition: The inverse sine or arcsine function.
For −1 ≤ x ≤ 1,
sin
1
x  y if, and only if, sin y  x
and 

2
 y

2
COMPOSITION OF SINE AND
INVERSE SINE
sin
1
sin x   x
for 

x

2
2
1
sin sin x  x for  1  x  1


DIFFERENTIATION OF
INVERSE SINE


d
1
1
sin x 
2
dx
1 x
1  x  1
THE INVERSE COSINE
FUNCTION
Definition: The inverse cosine or arccosine
function.
For −1 ≤ x ≤ 1,
cos 1 x  y if, and only if, cos y  x
and 0  y  
COMPOSITION OF COSINE AND
INVERSE COSINE
cos
1

cos x   x
1

for 0  x  
cos cos x  x for  1  x  1
DIFFERENTIATION OF
INVERSE COSINE


d
1
1
cos x  
2
dx
1 x
1  x  1
THE INVERSE TANGENT
FUNCTION
Definition: The inverse tangent or tangent
function.
For all real x,
1
tan x  y if, and only if, tan y  x
and 

2
 y

2
COMPOSITION OF TANGENT
AND INVERSE TANGENT
tan
1
tan x   x
for 

2
1
tan tan x  x for all x


x

2
DIFFERENTIATION OF
INVERSE TANGENT


d
1
1
tan x 
2
dx
1 x
THE INVERSE SECANT
FUNCTION
Definition: The inverse secant or arcsecant
function.
For | x | ≥ 1,
1
sec x  y if, and only if, sec y  x
and y  0,
3





,
2
2

COMPOSITION OF SECANT AND
INVERSE SECANT
sec sec x   x for x  0,
1


2
1

sec sec x  x for | x |  1
   , 
3
2
DIFFERENTIATION OF
INVERSE SINE


d
1
1
sec x 
2
dx
x x 1
| x | 1
The definitions for inverse cosecant and inverse
cotangent are on page 458 of the text.
The derivatives of inverse cosecant and inverse
cotangent are also on page 459 of the text.
ANTIDERIVATIVE FORMULAS
INVOLVING INVERSE TRIG.
FUNCTIONS
 1
1
dx  sin x  C

 1 x2
 1 dx  tan 1 x  C

 1 x2
1

1
dx

sec
xC

 x x2 1
GENERALIZED ANTIDERIVATIVE
FORMULAS INVOLVING
INVERSE TRIG. FUNCTIONS
1

1 x
dx  sin
C

a
 a2  x2
 1 dx  1 tan 1 x  C
 2
2
a x
a
a
1
1

1 x
dx  sec
C

a
a
 x x2  a2