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.
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include:
1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
Kamara
yes
Prince
how does the stomach protect itself from the damaging effects of HCl
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body
the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature