Math 151 Section 4.6
f ( x )  sin x on [
Inverse Trig Functions
 
, ] is one to one and has range [  1,1].
2 2
We write its inverse function, g(x)  sin 1 x  arcsin x.
Warning : The superscrip t in sin 1 x is not an exponent.
1
 (sin x ) 1  csc x are not the same as sin 1 x.
sin x
For this reason I use arcsin x.
arcsin x  y if and only if y is in [
 
, ] and sin y  x.
2 2
Evaluate each of the following : a )
c)
arcsin(sin (
2
)
3
d)
arcsin(cos
arcsin
1
2
b)
1
arcsin(  )
2
5
)
3
f ( x )  cos x is one to one on [ 0,  ] and has range [  1,1].
Its inverse function is g(x)  arccos x.
arccos x  y if and only if y is in [0,  ] and cos y  x.
Evaluate each : a )
c)
1
arccos  
2
 
arccos(cos   
 6
d)
b)
 1 
arccos  

2

cos(arccos (
3
)
2
Similarly we define arctan x and arc sec x

Range (arctan x )  ( 

Range ( arc sec x )  [ 0, )  ( ,  ]
2
2
Graph the functions a) y  arctan x and b) y  arc sec x .
Derivative s of Inverse Trig Functions
We derive the derivative of arcsin x. The others are similar.
For arcsin x  y , sin y  x. Using the chain rule we have
(cos y ) y '  1 .
f ( x)
arcsin x
arccos x
arctan x
arc sec x
y '
1
1

.
2
cos y
1 x
f '( x)
1
1  x2
1
1  x2
1
1  x2
1
x x2  1
 
, )
2 2
Examples
1.
Evaluate each expression .
a)
arctan(  1)
2.
Find the derivative of each function of x.
a)
f ( x )  x arctan x

arc sec  2
b)
f ( x )  arcsin( e x )
b)
3.
Evaluate each expression .
a)
1
cos(arcsin   )
4
b)

2
tan(arccos   )
5