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.
what's the difference between a firm and an industry
Abdul
firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅
Abdulraufu
Suppose the demand function that a firm faces shifted from
Qd 120 3P
to
Qd 90 3P
and the supply function has shifted from
QS
20 2P
to
QS
10 2P .
a) Find the effect of this change on price and quantity.
b) Which of the changes in demand and supply is higher?
Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods
Israr
Hi Sir please how do u calculate Cross elastic demand and income elastic demand?
Abari
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