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

 Page 1 / 2

## Problems in two dimensions

For the figure below, we are given that $BC=BD=x$ .

Show that $B{C}^{2}=2{x}^{2}\left(1+sin\theta \right)$ .

1. We want $CB$ , and we have $CD$ and $BD$ . If we could get the angle $B\stackrel{^}{D}C$ , then we could use the cosine rule to determine $BC$ . This is possible, as $▵ABD$ is a right-angled triangle. We know this from circle geometry, that any triangle circumscribed by a circle with one side going through the origin, is right-angled. As we have two angles of $▵ABD$ , we know $A\stackrel{^}{D}B$ and hence $B\stackrel{^}{D}C$ . Using the cosine rule, we can get $B{C}^{2}$ .

2. $A\stackrel{^}{D}B={180}^{\circ }-\theta -{90}^{\circ }={90}^{\circ }-\theta$

Thus

$\begin{array}{ccc}\hfill B\stackrel{^}{D}C& =& {180}^{\circ }-A\stackrel{^}{D}B\hfill \\ & =& {180}^{\circ }-\left({90}^{\circ }-\theta \right)\hfill \\ & =& {90}^{\circ }+\theta \hfill \end{array}$

Now the cosine rule gives

$\begin{array}{ccc}\hfill B{C}^{2}& =& C{D}^{2}+B{D}^{2}-2·CD·BD·cos\left(B\stackrel{^}{D}C\right)\hfill \\ & =& {x}^{2}+{x}^{2}-2·{x}^{2}·cos\left({90}^{\circ }+\theta \right)\hfill \\ & =& 2{x}^{2}+2{x}^{2}\left[\phantom{\rule{0.166667em}{0ex}},sin,\left({90}^{\circ }\right),cos,\left(\theta \right),+,sin,\left(\theta \right),cos,\left({90}^{\circ }\right)\right]\hfill \\ & =& 2{x}^{2}+2{x}^{2}\left[\phantom{\rule{0.166667em}{0ex}},1,·,cos,\left(,\theta ,\right),+,sin,\left(,\theta ,\right),·,0\right]\hfill \\ & =& 2{x}^{2}\left(1-sin\theta \right)\hfill \end{array}$
1. For the diagram on the right,
1. Find $A\stackrel{^}{O}C$ in terms of $\theta$ .
2. Find an expression for:
1. $cos\theta$
2. $sin\theta$
3. $sin2\theta$
3. Using the above, show that $sin2\theta =2sin\theta cos\theta$ .
4. Now do the same for $cos2\theta$ and $tan\theta$ .
2. $DC$ is a diameter of circle $O$ with radius $r$ . $CA=r$ , $AB=DE$ and $D\stackrel{^}{O}E=\theta$ . Show that $cos\theta =\frac{1}{4}$ .
3. The figure below shows a cyclic quadrilateral with $\frac{BC}{CD}=\frac{AD}{AB}$ .
1. Show that the area of the cyclic quadrilateral is $DC·DA·sin\stackrel{^}{D}$ .
2. Find expressions for $cos\stackrel{^}{D}$ and $cos\stackrel{^}{B}$ in terms of the quadrilateral sides.
3. Show that $2C{A}^{2}=C{D}^{2}+D{A}^{2}+A{B}^{2}+B{C}^{2}$ .
4. Suppose that $BC=10$ , $CD=15$ , $AD=4$ and $AB=6$ . Find $C{A}^{2}$ .
5. Find the angle $\stackrel{^}{D}$ using your expression for $cos\stackrel{^}{D}$ . Hence find the area of $ABCD$ .

## Problems in 3 dimensions

$D$ is the top of a tower of height $h$ . Its base is at $C$ . The triangle $ABC$ lies on the ground (a horizontal plane). If we have that $BC=b$ , $D\stackrel{^}{B}A=\alpha$ , $D\stackrel{^}{B}C=x$ and $D\stackrel{^}{C}B=\theta$ , show that

$h=\frac{bsin\alpha sinx}{sin\left(x+\theta \right)}$

1. We have that the triangle $ABD$ is right-angled. Thus we can relate the height $h$ with the angle $\alpha$ and either the length $BA$ or $BD$ (using sines or cosines). But we have two angles and a length for $▵BCD$ , and thus can work out all the remaining lengths and angles of this triangle. We can thus work out $BD$ .

2. We have that

$\begin{array}{ccc}\hfill \frac{h}{BD}& =& sin\alpha \hfill \\ \hfill ⇒\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}h& =& BDsin\alpha \hfill \end{array}$

Now we need $BD$ in terms of the given angles and length $b$ . Considering the triangle $BCD$ , we see that we can use the sine rule.

$\begin{array}{ccc}\hfill \frac{sin\theta }{BD}& =& \frac{sin\left(B\stackrel{^}{D}C\right)}{b}\hfill \\ \hfill ⇒\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}BD& =& \frac{bsin\theta }{sin\left(b\stackrel{^}{D}C\right)}\hfill \end{array}$

But $D\stackrel{^}{B}C={180}^{\circ }-\alpha -\theta$ , and

$\begin{array}{ccc}\hfill sin\left({180}^{\circ }-\alpha -\theta \right)& =& -sin\left(-\alpha -\theta \right)\hfill \\ & =& sin\left(\alpha +\theta \right)\hfill \end{array}$

So

$\begin{array}{ccc}\hfill BD& =& \frac{bsin\theta }{sin\left(D\stackrel{^}{B}C\right)}\hfill \\ & =& \frac{bsin\theta }{sin\left(\alpha +\theta \right)}\hfill \end{array}$
1. The line $BC$ represents a tall tower, with $C$ at its foot. Its angle of elevation from $D$ is $\theta$ . We are also given that $BA=AD=x$ .
1. Find the height of the tower $BC$ in terms of $x$ , $tan\theta$ and $cos2\alpha$ .
2. Find $BC$ if we are given that $x=140m$ , $\alpha ={21}^{\circ }$ and $\theta ={9}^{\circ }$ .

## Taxicab geometry

Taxicab geometry, considered by Hermann Minkowski in the 19th century, is a form of geometry in which the usual metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the (absolute) differences of their coordinates.

## Manhattan distance

The metric in taxi-cab geometry, is known as the Manhattan distance , between two points in an Euclidean space with fixed Cartesian coordinate system as the sum of the lengths of the projections of the line segment between the points onto the coordinate axes.

For example, the Manhattan distance between the point ${P}_{1}$ with coordinates $\left({x}_{1},{y}_{1}\right)$ and the point ${P}_{2}$ at $\left({x}_{2},{y}_{2}\right)$ is

#### Questions & Answers

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
hey
Giriraj
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!