131 Notes Section 3.6
Derivatives of Inverse trig functions.
Inverses of sinx and cosx:
∞A function has an inverse if it is one to one, but sinx and cosx are periodic and not one to one on the whole real line. So we restrict each to an interval where it is one to one.
f(x)=sinx is one to one on the interval [-π/2 , π/2] where it ranges from -1 at - π/2 to 1 at π/2 .
The inverse function to sin x then has domain = [ -1, 1] and
range= [-π/2 , π/2]. Recall that the domain of f is the range of f
and the range
of f is the domain of f . The inverse to sinx is
called sinx or arcsinx.
cos x is one to one on the interval [0, π] where it ranges from 1 at 0 to -1 at π.
The inverse to cosx is called cosx or arccosx and has domain
=[ -1, 1] and range=[0, π].
Find the derivatives
of arcsinx and arccosx:
If y = arcsin x then x= sin y. Differentiating x = sin y on both sides with respect to x, and using the chain rule,
* 1=(siny)' =(cosy)y' From the identity 
.
y is in the interval [-π/2 , π/2] which makes cosy positive, so 
.
Back to * we have 
Similary it can be shown that the derivative of arccos
x is 
.
Derivative of the Inverse
to tanx :
tanx is one to one on [-π/2 , π/2] and ranges from -∞ to ∞.
domain of arctanx =
(-∞,∞) and range of arctanx= (-π sec
y /2, π/2).
If y = arctanx then x=tany and 1=(secy)y' So y' = 1/ secy = 