<< Chapter < Page Chapter >> Page >
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

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where is the latest information on a no technology how can I find it
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now

Source:  OpenStax, Siyavula textbooks: grade 12 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11242/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Siyavula textbooks: grade 12 maths' conversation and receive update notifications?