Evaluating compositions of the form
f (
g−1 (
x ))
To evaluate compositions of the form
where
and
are any two of the functions sine, cosine, or tangent and
is any input in the domain of
we have exact formulas, such as
When we need to use them, we can derive these formulas by using the trigonometric relations between the angles and sides of a right triangle, together with the use of Pythagoras’s relation between the lengths of the sides. We can use the Pythagorean identity,
to solve for one when given the other. We can also use the
inverse trigonometric functions to find compositions involving algebraic expressions.
Evaluating the composition of a sine with an inverse cosine
Find an exact value for
Beginning with the inside, we can say there is some angle such that
which means
and we are looking for
We can use the Pythagorean identity to do this.
Since
is in quadrant I,
must be positive, so the solution is
See
[link] .
We know that the inverse cosine always gives an angle on the interval
so we know that the sine of that angle must be positive; therefore
Evaluating the composition of a sine with an inverse tangent
Find an exact value for
While we could use a similar technique as in
[link] , we will demonstrate a different technique here. From the inside, we know there is an angle such that
We can envision this as the opposite and adjacent sides on a right triangle, as shown in
[link] .
Using the Pythagorean Theorem, we can find the hypotenuse of this triangle.
Now, we can evaluate the sine of the angle as the opposite side divided by the hypotenuse.
Evolution models refer to mathematical and computational representations of the processes involved in biological evolution. These models aim to simulate and understand how species change over time through mechanisms such as natural selection, genetic drift, and mutation. Evolutionary models can be u
faruk
what are the modern trends in religious behaviours
shared standards of acceptable behavior by the group or appropriate behavior in a particular institution or those behaviors that are acceptable in a society
Lucius
that is how i understood it
Lucius
examples of societal norms
Diamond
Discuss the characteristics of the research located within positivist and the interpretivist paradigm