Evaluating compositions of the form
f (
f−1 (
y )) and
f−1 (
f (
x ))
For any trigonometric function,
for all
in the proper domain for the given function. This follows from the definition of the inverse and from the fact that the range of
was defined to be identical to the domain of
However, we have to be a little more careful with expressions of the form
Compositions of a trigonometric function and its inverse
Is it correct that
No. This equation is correct if
belongs to the restricted domain
but sine is defined for all real input values, and for
outside the restricted interval, the equation is not correct because its inverse always returns a value in
The situation is similar for cosine and tangent and their inverses. For example,
Given an expression of the form f
−1 (f(θ)) where
evaluate.
If
is in the restricted domain of
If not, then find an angle
within the restricted domain of
such that
Then
Evaluating compositions of the form
f−1 (
g (
x ))
Now that we can compose a trigonometric function with its inverse, we can explore how to evaluate a composition of a trigonometric function and the inverse of another trigonometric function. We will begin with compositions of the form
For special values of
we can exactly evaluate the inner function and then the outer, inverse function. However, we can find a more general approach by considering the relation between the two acute angles of a right triangle where one is
making the other
Consider the sine and cosine of each angle of the right triangle in
[link] .
Because
we have
if
If
is not in this domain, then we need to find another angle that has the same cosine as
and does belong to the restricted domain; we then subtract this angle from
Similarly,
so
if
These are just the function-cofunction relationships presented in another way.