# 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.

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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
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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Can someone give me problems that involes radical expressions like area,volume or motion of pendulum with solution