# Definitions, simple applications, and graphs of trigonometric  (Page 3/7)

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For most angles $\theta$ , it is very difficult to calculate the values of $sin\theta$ , $cos\theta$ and $tan\theta$ . One usually needs to use a calculator to do so. However, we saw in the above Activity that we could work these values out for some special angles. Some of these angles are listed in the table below, along with the values of the trigonometric functions at these angles. Remember that the lengths of the sides of a right angled triangle must obey Pythagoras' theorum. The square of the hypothenuse (side opposite the 90 degree angle) equals the sum of the squares of the two other sides.

 ${0}^{\circ }$ ${30}^{\circ }$ ${45}^{\circ }$ ${60}^{\circ }$ ${90}^{\circ }$ ${180}^{\circ }$ $cos\theta$ 1 $\frac{\sqrt{3}}{2}$ $\frac{1}{\sqrt{2}}$ $\frac{1}{2}$ 0 $-1$ $sin\theta$ 0 $\frac{1}{2}$ $\frac{1}{\sqrt{2}}$ $\frac{\sqrt{3}}{2}$ 1 0 $tan\theta$ 0 $\frac{1}{\sqrt{3}}$ 1 $\sqrt{3}$ $-$ 0

These values are useful when asked to solve a problem involving trig functions without using a calculator.

Find the length of x in the following triangle.

1. In this case you have an angle ( ${50}^{\circ }$ ), the opposite side and the hypotenuse.

So you should use $sin$

$sin{50}^{\circ }=\frac{x}{100}$
2. $⇒x=100×sin{50}^{\circ }$
3. Use the sin button on your calculator

$⇒x=76.6\mathrm{m}$

Find the value of $\theta$ in the following triangle.

1. In this case you have the opposite side and the hypotenuse to the angle $\theta$ .

So you should use $tan$

$tan\theta =\frac{50}{100}$
2. $⇒tan\theta =0.5$
3. Since you are finding the angle ,

use ${tan}^{-1}$ on your calculator

Don't forget to set your calculator to `deg' mode!

$⇒\theta =26.{6}^{\circ }$

The following videos provide a summary of what you have learnt so far.

## Finding lengths

Find the length of the sides marked with letters. Give answers correct to 2 decimal places.

## Simple applications of trigonometric functions

Trigonometry was probably invented in ancient civilisations to solve practical problems such as building construction and navigating by the stars. In this section we will show how trigonometry can be used to solve some other practical problems.

## Height and depth

One simple task is to find the height of a building by using trigonometry. We could just use a tape measure lowered from the roof, but this is impractical (and dangerous) for tall buildings. It is much more sensible to measure a distance along the ground and use trigonometry to find the height of the building.

[link] shows a building whose height we do not know. We have walked 100 m away from the building and measured the angle from the ground up to the top of the building. This angle is found to be $38,{7}^{\circ }$ . We call this angle the angle of elevation . As you can see from [link] , we now have a right-angled triangle. As we know the length of one side and an angle, we can calculate the height of the triangle, which is the height of the building we are trying to find.

If we examine the figure, we see that we have the opposite and the adjacent of the angle of elevation and we can write:

$\begin{array}{ccc}\hfill tan38,{7}^{\circ }& =& \frac{\mathrm{opposite}}{\mathrm{adjacent}}\hfill \\ & =& \frac{\mathrm{height}}{100\phantom{\rule{0.166667em}{0ex}}\mathrm{m}}\hfill \\ \hfill ⇒\mathrm{height}& =& 100\phantom{\rule{0.166667em}{0ex}}\mathrm{m}×tan38,{7}^{\circ }\hfill \\ & =& 80\phantom{\rule{0.166667em}{0ex}}\mathrm{m}\hfill \end{array}$

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
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