Goals
We will
review inverse functions,
develop psuedo-inverse functions for trigonometric functions,
and
calculate the derivatives of these inverse trig functions.
Inverse Functions
For a function to have an inverse function, it must be
one-to-one. That is each input produces exactly one output,
and each output is produced by exactly one input.
If a continuous function that has an inverse function goes up
can it come back down?
Give an alternate condition for a function to have a function
inverse.
Which of the six trig functions have inverse functions based
on this condition?
Invertible Trig Functions
For the sake of psuedo-trig inverse functions we define these
domain restricted versions.
π π
sin x
−2, 2
cos x
[0, π] tan x − π2 , π2
Derivative of arcsin x
y = arcsin x is the inverse of the restricted sine.
Re-write the above in terms of sine.
Take the derivative (remember x is the variable).
Use this to calculate the derivative of arcsin x.
Repeat this process to calculate the derivative of arccos x.
Anti-Derivatives
What is
R
What is
R
√ 1
dx?
1−x 2
√ 1
dx?
4−16x 2