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x 1 - x 2 + y 1 - y 2
Manhattan Distance (dotted and solid) compared to Euclidean Distance (dashed). In each case the Manhattan distance is 12 units, while the Euclidean distance is 36

The Manhattan distance changes if the coordinate system is rotated, but does not depend on the translation of the coordinate system or its reflection with respect to a coordinate axis.

Manhattan distance is also known as city block distance or taxi-cab distance. It is given these names because it is the shortest distance a car would drive in a city laid out in square blocks.

Taxicab geometry satisfies all of Euclid's axioms except for the side-angle-side axiom, as one can generate two triangles with two sides and the angle between them the same and have them not be congruent. In particular, the parallel postulate holds.

A circle in taxicab geometry consists of those points that are a fixed Manhattan distance from the center. These circles are squares whose sides make a 45 angle with the coordinate axes.

The great-circle distance is the shortest distance between any two points on the surface of a sphere measured along a path on the surface of the sphere (as opposed to going through the sphere's interior). Because spherical geometry is rather different from ordinary Euclidean geometry, the equations for distance take on a different form. The distance between two points in Euclidean space is the length of a straight line from one point to the other. On the sphere, however, there are no straight lines. In non-Euclidean geometry, straight lines are replaced with geodesics. Geodesics on the sphere are the great circles (circles on the sphere whose centers are coincident with the center of the sphere).The shape of the Earth more closely resembles a flattened spheroid with extreme values for the radius of curvature, or arcradius, of 6335.437 km at the equator (vertically) and 6399.592 km at the poles, and having an average great-circle radius of 6372.795 km.

Summary of the trigonomertic rules and identities

Pythagorean Identity Cofunction Identities Ratio Identities
cos 2 θ + sin 2 θ = 1 sin ( 90 - θ ) = cos θ tan θ = sin θ cos θ
cos ( 90 - θ ) = sin θ
Odd/Even Identities Periodicity Identities Double angle Identities
sin ( - θ ) = - sin θ sin ( θ ± 360 ) = sin θ sin ( 2 θ ) = 2 sin θ cos θ
cos ( - θ ) = cos θ cos ( θ ± 360 ) = cos θ cos ( 2 θ ) = cos 2 θ - sin 2 θ
tan ( - θ ) = - tan θ tan ( θ ± 180 ) = tan θ cos ( 2 θ ) = 2 cos 2 θ - 1
tan ( 2 θ ) = 2 tan θ 1 - tan 2 θ
Addition/Subtraction Identities Area Rule Cosine rule
sin ( θ + φ ) = sin θ cos φ + cos θ sin φ Area = 1 2 b c sin A a 2 = b 2 + c 2 - 2 b c cos A
sin ( θ - φ ) = sin θ cos φ - cos θ sin φ Area = 1 2 a b sin C b 2 = a 2 + c 2 - 2 a c cos B
cos ( θ + φ ) = cos θ cos φ - sin θ sin φ A r e a = 1 2 a c sin B c 2 = a 2 + b 2 - 2 a b cos C
cos ( θ - φ ) = cos θ cos φ + sin θ sin φ
tan ( θ + φ ) = tan φ + tan θ 1 - tan θ tan φ
tan ( θ - φ ) = tan φ - tan θ 1 + tan θ tan φ
Sine Rule
sin A a = sin B b = sin C c

End of chapter exercises

Do the following without using a calculator.

  1. Suppose cos θ = 0 . 7 . Find cos 2 θ and cos 4 θ .
  2. If sin θ = 4 7 , again find cos 2 θ and cos 4 θ .
  3. Work out the following:
    1. cos 15
    2. cos 75
    3. tan 105
    4. cos 15
    5. cos 3 cos 42 - sin 3 sin 42
    6. 1 - 2 sin 2 ( 22 . 5 )
  4. Solve the following equations:
    1. cos 3 θ · cos θ - sin 3 θ · sin θ = - 1 2
    2. 3 sin θ = 2 cos 2 θ
  5. Prove the following identities
    1. sin 3 θ = 3 sin θ - sin 3 θ 4
    2. cos 2 α ( 1 - tan 2 α ) = cos 2 α
    3. 4 sin θ · cos θ · cos 2 θ = sin 4 θ
    4. 4 cos 3 x - 3 cos x = cos 3 x
    5. tan y = sin 2 y cos 2 y + 1
  6. (Challenge question!) If a + b + c = 180 , prove that
    sin 3 a + sin 3 b + sin 3 c = 3 cos ( a / 2 ) cos ( b / 2 ) cos ( c / 2 ) + cos ( 3 a / 2 ) cos ( 3 b / 2 ) cos ( 3 c / 2 )

Questions & Answers

what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Siyavula textbooks: grade 12 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11242/1.2
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