# 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. Object on an inclined plane.

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. Depiction of a scenario associated with a forest fire.

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

We begin by identifying the angle θ in the figure below. Refined depiction of scenario.

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. Trigonometric representation of scenario.

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

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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
Devang Reply
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
Abhijith Reply
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?
s. Reply
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 ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
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
what is biological synthesis of nanoparticles
Sanket Reply
Can someone give me problems that involes radical expressions like area,volume or motion of pendulum with solution
BUGAL Reply

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Source:  OpenStax, Math 1508 (laboratory) engineering applications of precalculus. OpenStax CNX. Aug 24, 2011 Download for free at http://cnx.org/content/col11337/1.3
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