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

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$\left|{x}_{1},-,{x}_{2}\right|+\left|{y}_{1},-,{y}_{2}\right|$ 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 ${45}^{\circ }$ 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
${sin}^{3}a+{sin}^{3}b+{sin}^{3}c=3cos\left(a/2\right)cos\left(b/2\right)cos\left(c/2\right)+cos\left(3a/2\right)cos\left(3b/2\right)cos\left(3c/2\right)$

#### Questions & Answers

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Almas
are nano particles real
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Joseph
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no can't
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currently
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nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
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da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
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revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
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yes
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ya I also want to know the raman spectra
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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.
Damian
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Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
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Brian Reply
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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
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LITNING Reply
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LITNING Reply
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LITNING Reply
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LITNING
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Rafiq
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Rafiq
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Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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Bob Reply
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The nanotechnology is as new science, to scale nanometric
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nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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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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