If the two equal sides are of length
, then the hypotenuse,
, can be calculated as:
So, we have:
We can try something similar for
and
. We start with an equilateral triangle and we bisect one angle as shown in
[link] . This gives us the right-angled triangle that we need, with one angle of
and one angle of
.
An equilateral triangle with one angle bisected.
If the equal sides are of length
, then the base is
and the length of the vertical side,
, can be calculated as:
So, we have:
You do not have to memorise these identities if you know how to work them out.
Two useful triangles to remember
Alternate definition for
We know that
is defined as:
This can be written as:
But, we also know that
is defined as:
and that
is defined as:
Therefore, we can write
can also be defined as:
A trigonometric identity
One of the most useful results of the trigonometric functions is that they are related to each other. We have seen that
can be written in terms of
and
. Similarly, we shall show that:
We shall start by considering
,
We see that:
and
We also know from the Theorem of Pythagoras that:
So we can write:
Simplify using identities:
Prove:
Trigonometric identities
Simplify the following using the fundamental trigonometric identities:
Prove the following:
Reduction formula
Any trigonometric function whose argument is
,
,
and
(hence
) can be written simply in terms of
. For example, you may have noticed that the cosine graph is identical to the sine graph except for a phase shift of
. From this we may expect that
.
Function values of
Investigation : reduction formulae for function values of
Function Values of
In the figure P and P' lie on the circle with radius 2. OP makes an angle
with the
-axis. P thus has coordinates
. If P' is the reflection of P about the
-axis (or the line
), use symmetry to write down the coordinates of P'.
Write down values for
,
and
.
Using the coordinates for P' determine
,
and
.
From your results try and determine a relationship between the function values of
and
.
Function values of
In the figure P and P' lie on the circle with radius 2. OP makes an angle
with the
-axis. P thus has coordinates
. P' is the inversion of P through the origin (reflection about both the
- and
-axes) and lies at an angle of
with the
-axis. Write down the coordinates of P'.
Using the coordinates for P' determine
,
and
.
From your results try and determine a relationship between the function values of
and
.
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
When MP₁ becomes negative, TP start to decline.
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
ok
Shukri
how do you save a country economic situation when it's falling apart
Economic growth as an increase in the production and consumption of goods and services within an economy.but
Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
What do you think is more important to focus on when considering inequality ?
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
thank you so much 👍 sir
Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has
The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50.
A,Calculate quantities of x and y which maximize utility.
B,Calculate value of Lagrange multiplier.
C,Calculate quantities of X and Y consumed with a given price.
D,alculate optimum level of output .