# 0.13 Trigonometry

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Comment: Triangulation is a process that can be applied to solve problems in a number of areas of engineering including surveying, construction management, radar, sonar, lidar, etc.

## Refraction

Refraction is a physical phenomenon that occurs when light passes from one transparent medium (such as air) through another (for example, glass.) It is known that light travels at different speeds through different transparent media. The index of refraction of a medium is a measure of how much the speed of light is reduced as it passes through the medium. In the case of glass, the index of refraction is approximately 1.5. This means that light travels as a speed of $\frac{1}{1\text{.}5}=\frac{2}{3}$ times the speed of light in a vacuum.

Two common properties of transparent materials can be attributed to the index of refraction. One is that light rays change direction as they pass from one medium through another. Secondly, light is partially reflected when it passes from one medium to another medium with a different index of refraction. We will focus on the first of these properties in this reading.

In optics, which is a field of physics, you will learn about Snell's law, which is also known as Descartes' law after the scientist, Rene Descartes . Snell’s law takes the form of an equation that states the relationship between the angle of incidence and the angle of refraction for light passing from one medium to another. Stated mathematically, Snell’s law is

$\frac{\text{sin}\left({\theta }_{\text{incidence}}\right)}{\text{sin}\left({\theta }_{\text{refraction}}\right)}=\frac{{c}_{\text{incidence}}}{{c}_{\text{refraction}}}$

It follows that

$\frac{\text{sin}\left({\theta }_{\text{incidence}}\right)}{\text{sin}\left({\theta }_{\text{refraction}}\right)}=\frac{{I}_{2}}{{I}_{1}}$

where ${I}_{1}\text{and}{I}_{2}$ are the Index of Refraction of medium 1 and medium 2 respectively.

Consider a situation where light rays pass are shined from air through a tank of water. This situation is illustrated below.

The Index of refraction for air is 1.0003 and that of water is 1.3000. Let us assume that the angle that light enters the water is 21 0 40’, what is the angle of refraction, w ?

From Snell’s law, we know

$\frac{{I}_{W}}{{I}_{A}}=\frac{\text{sin}\left(a\right)}{\text{sin}\left(w\right)}$
$\text{sin}\left(w\right)=\frac{{I}_{A}}{{I}_{W}}\text{sin}\left(a\right)$

$\text{sin}\left(w\right)=\frac{{I}_{A}}{{I}_{W}}\text{sin}\left(a\right)$

Substituting in the numerical values for I A , I W and a yield

$\text{sin}\left(w\right)=0\text{.}\text{2841}$

We now make use of the inverse sine function

$w={\text{sin}}^{-1}\left(0\text{.}\text{2841}\right)$

$w={\text{16}}^{0}{\text{30}}^{\text{'}}$

We conclude that the refracted ray will travel through the water at an angle of refraction of 16 0 30’ .

## Exercises

1. A 50 ft ladder leans against the top of a building which is 30 ft tall. Determine the angle the ladder makes with the horizontal. Also determine the distance from the base of the ladder to the building.
2. A straight trail leads from the Alpine Hotel at elevation 8,000 feet to a scenic overlook at elevation 11,100 feet. The length of the trail is 14,100 feet. What is the inclination angle β in degrees? What is the value of β in radians?
3. A ray of light moves from a media whose index of refraction is 1.200 to another whose index of refraction is 1.450. The angle of incidence of the ray as it intersects the interface of the two media is 15 0 . Sketch the geometry of the situation and determine the value of the angle of refraction.
4. One-link planar robots can be used to place pick up and place parts on work table. A one-link planar robot consists of an arm that is attached to a work table at one end. The other end is left free to rotate about the work space. If l = 5 cm, sketch the position of the robot and determine the ( x, y ) coordinates of point p ( x , y ) for the following values for θ: (50˚, 2π/3 rad, -20˚, and -5 π/4 rad).

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
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
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
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
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?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
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Good
Can someone give me problems that involes radical expressions like area,volume or motion of pendulum with solution