The other cases can be proved in an identical manner.
Find
:
The cosine rule
Solve the following triangles
i.e. find all unknown sides and angles
ABC in which
;
and
XYZ in which
;
and
RST in which RS
; ST
and RT
KLM in which KL
; LM
and KM
JHK in which
; JH
and HK
DEF in which
;
and
Find the length of the third side of the
XYZ where:
;
km and
km
;
cm;
and
cm
Determine the largest angle in:
JHK in which JH
; HK
and JK
PQR where
;
and
The area rule
The Area Rule
The area rule applies to any triangle and states that the area of a triangle is given by half the product of any two sides with the sine of the angle between them.
That means that in the
, the area is given by:
In order show that this is true for all triangles, consider
.
The area of any triangle is half the product of the base and the perpendicular height. For
, this is:
However,
can be written in terms of
as:
So, the area of
is:
Using an identical method, the area rule can be shown for the other two angles.
Find the area of
ABC:
ABC is isosceles, therefore AB
AC
and
. Hence
. Now we can use the area rule to find the area:
The area rule
Draw sketches of the figures you use in this exercise.
Find the area of
PQR in which:
;
and
;
and
;
and
Find the area of:
XYZ with XY
cm; XZ
cm and
PQR with PR
cm; PQ
cm and
EFG with FG
cm; EG
cm and
Determine the area of a parallelogram in which two adjacent sides are 10 cm and 13 cm and the angle between them is
.
If the area of
ABC is 5000 m
with
m and
m, what are the two possible sizes of
?
Summary of the trigonometric rules and identities
Pythagorean Identity
Ratio Identity
Odd/Even Identities
Periodicity Identities
Cofunction Identities
Sine Rule
Area Rule
Cosine Rule
Exercises
Q is a ship at a point 10 km due South of another ship P. R is a lighthouse on the coast such that
.
Determine:
the distance QR
the shortest distance from the lighthouse to the line joining the two ships (PQ).
WXYZ is a trapezium (WX
XZ) with WX
m; YZ
m;
and
Determine the distances XZ and XY.
Find the angle
.
On a flight from Johannesburg to Cape Town, the pilot discovers that he has been flying
off course. At this point the plane is 500 km from Johannesburg. The direct distance between Cape Town and Johannesburg airports is 1 552 km. Determine, to the nearest km:
The distance the plane has to travel to get to Cape Town and hence the extra distance that the plane has had to travel due to the pilot's error.
The correction, to one hundredth of a degree, to the plane's heading (or direction).
ABCD is a trapezium (ie. AB
CD). AB
;
;
and
.
Find an expression for the length of CD in terms of
,
,
and
.
A surveyor is trying to determine the distance between points X and Z. However the distance cannot be determined directly as a ridge lies between the two points. From a point Y which is equidistant from X and Z, he measures the angle
.
If XY
and
, show that XZ
.
Calculate XZ (to the nearest kilometre) if
km and
.
Find the area of WXYZ (to two decimal places):
Find the area of the shaded triangle in terms 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 .