# 8.1 Applications of trig functions (2d & 3d), other geometries  (Page 2/2)

 Page 2 / 2
$\left|{x}_{1},-,{x}_{2}\right|+\left|{y}_{1},-,{y}_{2}\right|$

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 ${45}^{\circ }$ 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 ${cos}^{2}\theta +{sin}^{2}\theta =1$ $sin\left({90}^{\circ }-\theta \right)=cos\theta$ $tan\theta =\frac{sin\theta }{cos\theta }$ $cos\left({90}^{\circ }-\theta \right)=sin\theta$ Odd/Even Identities Periodicity Identities Double angle Identities $sin\left(-\theta \right)=-sin\theta$ $sin\left(\theta ±{360}^{\circ }\right)=sin\theta$ $sin\left(2\theta \right)=2sin\theta cos\theta$ $cos\left(-\theta \right)=cos\theta$ $cos\left(\theta ±{360}^{\circ }\right)=cos\theta$ $cos\left(2\theta \right)={cos}^{2}\theta -{sin}^{2}\theta$ $tan\left(-\theta \right)=-tan\theta$ $tan\left(\theta ±{180}^{\circ }\right)=tan\theta$ $cos\left(2\theta \right)=2{cos}^{2}\theta -1$ $tan\left(2\theta \right)=\frac{2tan\theta }{1-{tan}^{2}\theta }$ Addition/Subtraction Identities Area Rule Cosine rule $sin\left(\theta +\phi \right)=sin\theta cos\phi +cos\theta sin\phi$ $\mathrm{Area}=\frac{1}{2}bcsinA$ ${a}^{2}={b}^{2}+{c}^{2}-2bccosA$ $sin\left(\theta -\phi \right)=sin\theta cos\phi -cos\theta sin\phi$ $\mathrm{Area}=\frac{1}{2}absinC$ ${b}^{2}={a}^{2}+{c}^{2}-2accosB$ $cos\left(\theta +\phi \right)=cos\theta cos\phi -sin\theta sin\phi$ $Area=\frac{1}{2}acsinB$ ${c}^{2}={a}^{2}+{b}^{2}-2abcosC$ $cos\left(\theta -\phi \right)=cos\theta cos\phi +sin\theta sin\phi$ $tan\left(\theta +\phi \right)=\frac{tan\phi +tan\theta }{1-tan\theta tan\phi }$ $tan\left(\theta -\phi \right)=\frac{tan\phi -tan\theta }{1+tan\theta tan\phi }$ Sine Rule $\frac{sinA}{a}=\frac{sinB}{b}=\frac{sinC}{c}$

## End of chapter exercises

Do the following without using a calculator.

1. Suppose $cos\theta =0.7$ . Find $cos2\theta$ and $cos4\theta$ .
2. If $sin\theta =\frac{4}{7}$ , again find $cos2\theta$ and $cos4\theta$ .
3. Work out the following:
1. $cos{15}^{\circ }$
2. $cos{75}^{\circ }$
3. $tan{105}^{\circ }$
4. $cos{15}^{\circ }$
5. $cos{3}^{\circ }cos{42}^{\circ }-sin{3}^{\circ }sin{42}^{\circ }$
6. $1-2{sin}^{2}\left(22.{5}^{\circ }\right)$
4. Solve the following equations:
1. $cos3\theta ·cos\theta -sin3\theta ·sin\theta =-\frac{1}{2}$
2. $3sin\theta =2{cos}^{2}\theta$
5. Prove the following identities
1. ${sin}^{3}\theta =\frac{3sin\theta -sin3\theta }{4}$
2. ${cos}^{2}\alpha \left(1-{tan}^{2}\alpha \right)=cos2\alpha$
3. $4sin\theta ·cos\theta ·cos2\theta =sin4\theta$
4. $4{cos}^{3}x-3cosx=cos3x$
5. $tany=\frac{sin2y}{cos2y+1}$
6. (Challenge question!) If $a+b+c={180}^{\circ }$ , prove that
${sin}^{3}a+{sin}^{3}b+{sin}^{3}c=3cos\left(a/2\right)cos\left(b/2\right)cos\left(c/2\right)+cos\left(3a/2\right)cos\left(3b/2\right)cos\left(3c/2\right)$

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
Got questions? Join the online conversation and get instant answers!