# 0.13 Trigonometry

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$\text{sin}\left({3}^{0}\right)=\frac{\text{31},\text{680}}{C}$

Rearranging terms we find

$C=\frac{\text{31},\text{680}}{\text{sin}\left({3}^{o}\right)}$

Calculation and rounding to 3 significant digits yields the result

$C=\text{605},\text{320}\text{ft}$

Let us ponder a second question based upon the data presented in the original problem.

Question 2 : What is the ground distance traveled by the airplane as it moves from its departure point to its cruise altitude?

Solution : Referring to Figure 2, we observe that we must find the length of the adjacent side in order to answer the question. We can use the definition of the tangent to guide our solution.

$\text{tan}\left({3}^{0}\right)=\frac{\text{Opposite}\text{side}}{\text{Adjacent}\text{side}}$

Denoting the adjacent side by the symbol A , we obtain

$A=\frac{\text{Opposite}\text{side}}{\text{tan}\left({3}^{0}\right)}$
$A=\frac{\text{31},\text{680}}{0\text{.}\text{0524}}$

After rounding to 3 significant digits, we obtain the solution

$A=\text{604},\text{580}\text{ft}$

## Inclined plane

Work is an important concept in virtually every field of science and engineering. It takes work to move an object; it takes work to move an electron through an electric field; it takes work to overcome the force of gravity; etc.

Let’s consider the case where we use an inclined plane to assist in the raising of a 300 pound weight. The inclined plane situated such that one end rests on the ground and the other end rests upon a surface 4 feet aove the ground. This situation is depicted in Figure 3.

Question 3: Suppose that the length of the inclined plane is 12 feet. What is the angle that the plane makes with the ground?

Clearly, the length of the inclined plane is same as that of the hypotenuse shown in the figure. Thus, we may use the sine function to solve for the angle

$\text{sin}\left(\theta \right)=\frac{4}{\text{12}}=0\text{.}\text{333}$

In order to solve for the angle, we must make use of the inverse sine function as shown below

${\text{sin}}^{-1}\left(\text{sin}\left(\theta \right)\right)={\text{sin}}^{-1}\left(0\text{.}\text{333}\right)$
$\theta =\text{19}\text{.}{\text{45}}^{0}$

So we conclude that the inclined plane makes a 19.85 0 angle with the ground.

Neglecting any effects of friction, we wish to determine the amount of work that is expended in moving the block a distance ( L ) along the surface of the inclined plane.

## Surveying

Let us now turn our attention to an example in the field of surveying. In particular, we will investigate how trigonometry can be used to help forest rangers combat fires. Let us suppose that a fire guard observes a fire due south of her Hilltop Lookout location. A second fire guard is on duty at a Watch Tower that is located 11 miles due east of the Hilltop Lookout location. This second guard spots the same fire and measures the bearing (angle) at 215 0 from North. The figure below illustrates the geometry of the situation.

Question: How far away is the fire from the Hilltop Lookout location?

We begin by identifying the angle θ in the figure below.

The value of θ can be found via the equation

$\theta ={\text{90}}^{0}-{\text{35}}^{0}$
$\theta ={\text{55}}^{0}$

So we can simplify the drawing as shown below.

Our problem reduces to solving for the value of b .

$\text{tan}\left({\text{55}}^{0}\right)=\frac{b}{\text{11}\text{miles}}$
$b=\left(1\text{.}\text{43}\right)\left(\text{11}\text{miles}\right)=\text{15}\text{.}7\text{miles}$

We conclude that the fire is located 15.7 miles south of the Hilltop Lookout location.

#### 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
ya I also want to know the raman spectra
Bhagvanji
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
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Good
Can someone give me problems that involes radical expressions like area,volume or motion of pendulum with solution