Satellite navigation systems

Satellite navigation systems need to know the precise distances between points. One way of measuring the distance between two points is simply to walk, drive or fly from one to the other and measure it. But this can be time-consuming and expensive, especially if the points you’re interested in lie far out in the ocean or even in space. Using a triangle is one way of getting around this problem: call the point where you (or a satellite) are observing from A and put another observer at a point B, which is a known distance d away from you. Suppose you want to know how far a third point, called C, is away from you. The points A, B and C form a triangle. You can measure the angle a between the straight line from A to C and the straight line from A to B, and tell the other observer to measure the corresponding angle b at her end. Now you know two angles of the triangle and one side, the A to B of length d, and so you can calculate the other sides and angles, in particular the distance between you and C.

These kinds of techniques are used not only in satellite navigation, but in all kinds of navigation involving cars, ships, planes and space craft. They are used in geography, land surveying and cartography (map making). They are especially important in astronomy to measure the distance between Earth and the most distant stars.

Music
You may know from school that the graphs of the functions sin(x) and cos(x) look like waves. Sound travels in waves, although these are not necessarily as regular as those of the sine and cosine functions. However, a few hundred years ago, mathematicians realised that any wave at all is made up of sine and cosine waves. This fact lies at the heart of computer music. Since a computer cannot listen to music as we do, the only way to get music into a computer is to represent it mathematically by its constituent sound waves. This is why sound engineers, those who research and develop the newest advances in computer music technology, and sometimes even composers have to understand the basic laws of trigonometry.

But all of this doesn’t only apply to sound. Waves move across the oceans, earthquakes produce shock waves and light can be thought of as travelling in waves. This is why trigonometry is also used in oceanography, seismology, optics and many other fields like meteorology and the physical sciences.

Architecture

Many modern buildings have beautifully curved surfaces. Making these curves out of steel, stone, concrete or glass is extremely difficult, if not impossible. One way around this problem is to piece the surface together out of many flat panels, each sitting at an angle to the one next to it, so that all together they create what looks like a curved surface. The more regular these shapes, the easier the building process. Regular flat shapes like squares, pentagons and hexagons, can be made out of triangles, and so trigonometry plays an important role in architecture.

Digital imaging

How can a computer generate complex images? In theory, the computer needs an infinite amount of information to do this: it needs to know the precise location and colour of each of the infinitely many points on the image to be produced. In practise, this is of course impossible, a computer can only store a finite amount of information. To make the image as detailed and accurate as possible, computer graphic designers resort to a technique called triangulation. As in the architecture example above, they approximate the image by a large number of triangles, so the computer only needs to store a finite amount of data. The edges of these triangles form what looks like a wire frame of the object in the image. Using this wire frame, it is also possible to make the object move realistically.

Digital imaging is also used extensively in medicine, for example in CAT and MIR scans. Again, triangulation is used to build accurate images from a finite amount of information. It is also used to build “maps” of things like tumours, which help decide how x-rays should be fired at it in order to destroy it.

Trigonometric functions such as sine, cosine, and tangent are used in computations in trigonometry. These functions relate measurements of angles to measurements of associated straight lines as described later in this short course.

Trig functions are not easy to compute like polynomials are. So much time goes into computing them in ancient times that tables were made for their values. Even with tables, using trig functions takes time because any use of a trig function involves at least one multiplication or division, and, when several digits are involved, even multiplication and division are slow. In the early 17th century computation sped up with the invention of logarithms and soon after slide rules. With the advent of calculators computation has become easy. Tables, logarithms, and slide rules aren’t needed in trigonometric computations. All you have to do is enter the numbers and push a few buttons to get the answer. One of the things that used to make learning trig difficult was performing the computations.

Usage

Trigonometry has an enormous variety of applications. The ones mentioned explicitly in textbooks and courses on trigonometry are its uses in practical endeavors such as navigation, land surveying, building, and the like. It is also used extensively in a number of academic fields, primarily mathematics, science and engineering.
Among the lay public of non-mathematicians and non-scientists, trigonometry is known chiefly for its application to measurement problems, yet is also often used in ways that are far more subtle, such as its place in the theory of music; still other uses are more technical, such as in number theory. The mathematical topics of Fourier series and Fourier transforms rely heavily on knowledge of trigonometric functions and find application in a number of areas, including statistics.

History

By the 10th century, in the work of Abū al-Wafā’ al-Būzjānī, Muslim mathematicians were using all six trigonometric functions, after discovering the secant, cotangent and cosecant functions. Abu al-Wafa had sine tables in 0.25° increments, to 8 decimal places of accuracy, and accurate tables of tangent values. He also developed the following trigonometric formula:

Also in the 10th century, Al-Battani was responsible for establishing a number of important trigometrical relationships such as:

Functions

Function Abbreviation Identities (using radians)
Sine sin
Cosine cos
Tangent tan(or tg)
Cosecant csc(or cosec)
Secant sec
Cotangent cot(or ctg or ctn)

In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. They are important in the study of triangles and modeling periodic phenomena, among many other applications. Trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers.
In modern usage, there are six basic trigonometric functions, which are tabulated here along with equations relating them to one another. Especially in the case of the last four, these relations are often taken as the definitions of those functions, but one can define them equally well geometrically or by other means and then derive these relations.
What can you do with trig?(App)

Historically, it was developed for astronomy and geography, but scientists have been using it for centuries for other purposes, too. Besides other fields of mathematics, trig is used in physics, engineering, and chemistry. Within mathematics, trig is used in primarily in calculus (which is perhaps its greatest application), linear algebra, and statistics. Since these fields are used throughout the natural and social sciences, trig is a very useful subject to know.
Astronomy and geography
Trigonometric tables were created over two thousand years ago for computations in astronomy. The stars were thought to be fixed on a crystal sphere of great size, and that model was perfect for practical purposes. Only the planets moved on the sphere. (At the time there were seven recognized planets: Mercury, Venus, Mars, Jupiter, Saturn, the moon, and the sun. Those are the planets that we name our days of the week after. The earth wasn’t yet considered to be a planet since it was the center of the universe, and the outer planets weren’t discovered then.) The kind of trigonometry needed to understand positions on a sphere is called spherical trigonometry. Spherical trigonometry is rarely taught now since its job has been taken over by linear algebra. Nonetheless, one application of trigonometry is astronomy.

As the earth is also a sphere, trigonometry is used in geography and in navigation. Ptolemy (100-178) used trigonometry in his Geography and used trigonometric tables in his works. Columbus carried a copy of Regiomontanus’ Ephemerides Astronomicae on his trips to the New World and used it to his advantage.
Engineering and physics
Although trigonometry was first applied to spheres, it has had greater application to planes. Surveyors have used trigonometry for centuries. Engineers, both military engineers and otherwise, have used trigonometry nearly as long.

Physics lays heavy demands on trigonometry. Optics and statics are two early fields of physics that use trigonometry, but all branches of physics use trigonometry since trigonometry aids in understanding space. Related fields such as physical chemistry naturally use trig.
Mathematics and its applications
Of course, trigonometry is used throughout mathematics, and, since mathematics is applied throughout the natural and social sciences, trigonometry has many applications. Calculus, linear algebra, and statistics, in particular, use trigonometry and have many applications in the all the sciences.

1. Heavy metals
a. Sources
i. Car and truck exhaust, worn tires and engine parts, brake linings, weathered paint, and rust
b. Effects
i. Toxic to aquatic life and can potentially contaminate groundwater
2. Industry and power plant discharge
a. Sources
i. Litter washed into storm drains and creeks, chemicals
b. Effects
i. Looks and smells unpleasant; harms wildlife
ii. Toxins and metals absorbed in various aquatic life cause medical problems in humans when they consume contaminated fish and shellfish
3. Landscape pollutants
a. Sources
i. Fertilizers, weed killer, insecticide, fungicides, and grass, tree and shrub clippings wash to storm drains or soak into groundwater when it rains
b. Effects
· Phosphorus and nitrogen from fertilizers cause algal blooms, which depletes water of oxygen, killing fish and aquatic life
· Pesticides and herbicides can be harmful to humans and aquatic organisms (some are carcinogenic or attack the nervous system)
· Loose grass clippings and leaves clog drainage systems and/or cause algal blooms in water
4. Automobile pollutants
a. Sources
· Oil, antifreeze, brake fluid, grease and metals on streets and driveways run off pavement to stormdrains or soak into groundwater
· Nitrogen and other contaminants emitted from automobiles settle in water
· Oil, grease, transmission fluids, etc. spilled from automobiles, trucks, buses, planes, etc. wash to storm drain or creek
b. Effects
· Oil, petroleum products and other toxins from automobiles kill fish, plants, aquatic life and even people (contaminate drinking water). Used oil from a single oil change can ruin a million gallons of water-a year’s supply for 50 people.
· Some of these toxins and metals are absorbed in various aquatic life and can cause medical problems to humans when contaminated fish and shellfish are consumed.
· Pollutants such as heavy metals and automobile fluids are toxic to aquatic life (interferes with photosynthesis, respiration, growth and reproduction).

5. Organic waste
a. Sources
· Failing sewer systems spilling out raw sewage after a heavy rain
· Leaking or failing septic systems
· Pet wastes not collected and disposed of appropriately
· Pathogens from rotting food or dead animals
· Discharge from food-processing plants, meat-packing houses, dairies and other industrial sources
· Organic waste from fibers originating from textile and plant processing plants
· Wastewater treatment plants
b. Effects
· Fecal coliform bacteria in pet droppings and septic tank overflows can cause infections and diseases by getting into drinking water and recreation areas
· Pathogens from food and dead animals may also cause infections and diseases if they enter water sources
· Phosphorus and nitrogen from organic material cause algal blooms, which depletes water of oxygen, killing fish and aquatic life
6. Household chemicals
a. Sources
· Improperly disposed paint, solvents and other chemicals runoff or soak into the ground
· Household and commercial cleaning agents wash into water and stormdrains
· Washing car
b. Effects
· Paint, cleaning supplies and other toxic materials contaminate drinking water and kill fish, animals and plants
· Detergents cause explosive plant and algae growth, which depletes water of oxygen, killing fish and animals as well as creating a terrible smell
7. Thermal pollution
a. Sources
· Discharge of heated water from power plants
· Removal of shade trees along creek banks
b. Effects
· Inhibits fish growth and reproduction and can be fatal to aquatic life
· Increases evaporation thus decreasing flow rate of river

Consequences of Pollution

Overuse and pollution of the world’s freshwater resources are a recent development. Their long-term consequences are unknown. Already, however, they have taken a heavy toll on the environment, and they pose increasing risks for many species. Polluted water and lack of sanitation also are fostering a human health tragedy. Moreover, the sad state of freshwater resources contributes to the deterioration of coastal waters and seas.
In 1996 the world’s human population was using an estimated 54% of all the accessible freshwater contained in rivers, lakes, and underground aquifers. This percentage is conservatively projected to climb to at least 70% by 2025, reflecting population growth alone, and by much more if per capita consumption continues to rise at its current pace. As humankind withdraws a growing share of all water, less remains to maintain the vital ecosystems on which we also depend.
A substantial portion of the total freshwater available in the hydrological cycle is needed to sustain natural aquatic ecosystems—marshes, rivers, coastal wetlands—and the millions of species that they shelter. Healthy natural ecosystems are indispensable regulators of water quality and quantity. For example, flood plain wetlands soak up and store water when rivers flood their banks, reducing downstream damage.
The value of these environmental services to humankind is immense. One estimate, made by Robert Costanza, director of the Institute of Ecological Economics at the University of Maryland, puts the global value of wetlands at close to US$5 trillion dollars a year, based on their value as flood regulators, waste treatment plants, and wildlife habitats, as well as for fisheries production and recreation, among other uses (92). New York City is spending US$1 billion to conserve and protect water catchment areas in upstate New York—the source of the city’s drinking water. The alternative would be to spend $5 billion on a state-of-the-art water filtration plant that would cost an additional $300 million a year to operate.
In virtually all regions of the world, careless use of water resources is harming the natural environment. Globally, over 20% of all freshwater fish species are either endangered or vulnerable or recently have been made extinct. As the following examples demonstrate, overusing and misusing freshwater resources carries serious consequences for natural species as well as for human populations:
· Diverting water from the Nile River, along with build-up of sediments trapped behind dams and barrages, has caused the fertile Nile delta to shrink. Of 47 commercial species of fish, about 30 have become extinct or virtually extinct. Delta fisheries that once supported over a million people have been wiped out.
· Lake Chad, in Africa’s Sahel region, has shrunk from 25,000 square kilometers to just 2,000 square kilometers in the last three decades from periodic droughts and massive diversions of water for irrigation. The lake’s once rich fisheries have entirely collapsed .
· Despite cleanup efforts, the Rhine River, which runs through the industrial heartland of Western Europe, is so polluted that it has lost 8 of its 44 species of fish. Another 25 species have become rare or are endangered.
· In Colombia fish production in the Magdalena River plunged from 72,000 metric tons in 1977 to 23,000 metric tons by 1992—a two-thirds drop in 15 years—as a result of agricultural, urban, and industrial development and deforestation in the river’s watershed.
· Southeast Asia’s Mekong River has had a two-thirds drop in fisheries production due to dams, deforestation, and conversion of 1,000 square kilometers of mangrove swamps into rice paddies and fish ponds.
· The US state of California has lost over 90% of its wetlands. As a result, nearly two-thirds of the state’s native fish are extinct, endangered, threatened, or in decline
Case Study

The hidden impacts of air pollution on the poor: a case
study of heavy metal contamination of vegetables in
Indian cities

Vegetable crops are often grown in polluted and degraded environmental conditions
in the peri-urban (or urban fringe) zone and are subject to further pollution from vehicles
and industries during marketing. There is therefore significant cause for concern
regarding the potential impacts of air pollution on crop yield and quality.
Levels of contamination of Cu, Zn, Cd and Pb were measured in spinach beet (palak),
cauliflower and okra at market and field sites in Varanasi, and in palak in Delhi, and
were compared with national (Indian) and International permissible limits. In Varanasi
markets, the mean heavy metal contamination levels significantly exceeded the Indian
prevention of food adulteration act (PFA) limits for Cd, Cu and Zn for much of the
year for all three crops. Pb did not exceed the PFA limits, but the majority of samples
did exceed the more stringent EU or CODEX permissible limits. In Delhi markets
the majority of palak samples contained Pb concentrations that exceeded the Indian
PFA limit. The considerably lower contamination levels measured in crops at field
production sites indicates that a significant proportion of the contamination occurs
during transport to market or at the point of sale. Heavy metal contamination could be
reduced, often to below PFA permissible limits, by twice washing in clean water.
This paper is one of a series of outputs from a major interdisciplinary research project
carried out to assess the nature and significance of aerial deposition of heavy metals
on the safety of vegetables consumed in urban India (with particular emphasis on
impacts on the poor); to explore appropriate technical and institutional measures to
address the issue, and to draw lessons for policy approaches to improve food safety
in India. The study brought together a cross-sectoral team to develop new types of
partnerships and new ways of working, in order to understand and address the impacts
of newly emerging environmental threats to the food system on the livelihoods of the
poor. The study is a pointer to the inefficacy of current policy approaches towards
ensuring safety of food to the consumer. Current policy relates to food standards,
environmental standards, industrial siting, peri-urban agriculture and consumer rights
separately and is inadequate to tackle the issue comprehensively. Whilst progress is
being made with the proposed new integrated food safety bill, there is still no emphasis
on fresh produce rather than processed food, or recognition of environmental pollution
as a threat to food safety.

Pollution from landfill

Waste management places considerable strain on the environment. Waste disposal sites, whether active or closed, can result in serious pollution of groundwater due to leachate. A recent survey by the Environment Agency of groundwater pollution in England and Wales, revealed that the main sources of pollution (in order) are:
1. Landfill leachate 2. Chemical and metal processing industries 3. Gas works 4. Power stations 5. Petrol service stations
Landfill leachates rank alongside heavy metals and organic compounds as the most frequently recorded pollutants. Locally, in the Great Stour catchment, 26 sites of waste disposal have been identified by the Environment Agency as potential groundwater contaminants. Brett Waste Management Ltd. (Shelford Quarry, Broad Oak Road) is one of them. At present, however, there is no contamination of the river via groundwater leakage of leachate from any of these sites; all landfills are under licence, and are monitored closely. There are alternative ways of disposing of our domestic, commercial and industrial solid waste, but for the moment, landfill remains the cheapest disposal option.
potential pollution from landfill waste disposal
Solid household waste is recognised as a major threat to the environment, with high pollution potential. The average household in Britain generates 600 kg of waste per year (11.5 kg per week); the total waste generated nationally is 380 million tonnes per year! Each household produces on average each week:
3 kg paper 1.25kg glass 2 kg cans 1 kg plastics
In mixed (unseparated) compostable waste, including kitchen and garden refuse, there is also a diverse range of other materials, some of which are potentially hazardous. These hazardous substances include: decorating products (paints, stains, varnish, paint thinners), garden products (pesticides, fungicides, herbicides), vehicle products (engine oil, brake fluid, antifreeze, car batteries), household cleaners (bleach, disinfectant, air fresheners), toiletries (cosmetics, old medicines) and other miscellaneous items. Batteries from watches, radios, mobile phones, etc. may contain heavy metals like mercury, nickel, cadmium.
When such household waste in landfill sites is acted on by rainwater, the organic and inorganic constituents are dissolved, and a highly toxic leachate results, collecting at the base of the landfill. This is normally high in heavy metals, ammonia, toxic organic compounds and pathogens. It also has a high BOD, and if it escapes into the groundwater serious contamination results. Meanwhile, at the top of the landfill, gas is produced by the fermentation of organic material. Approximately equal quantities of carbon dioxide (CO 2 ) and methane (CH 4 ) are released. Both are greenhouse gases, but methane is 26 times more effective than carbon dioxide in this respect. In addition to leachate and biogas problems, landfill sites are very unpopular with local residents: traffic, smell, noise, vermin, seagulls, blown litter, and disease can all spoil the neighbourhood and lower property prices.
Preventative measures to limit pollution from Landfill waste
In the past it was considered acceptable to allow leachate to seep away slowly and be dispersed through the ground. This old dilute-and-disperse method of waste disposal is now no longer acceptable. New methods are based on the idea of containment. Landfills are lined with clay and flexible synthetic membranes intended to prevent leachate escaping and contaminating the groundwater. Leachate is drained through a horizontal array of perforated pipes to be collected by a sump for treatment. Methane (biogas) is also collected at the top, and is either vented to the air or tapped off and re-cycled for industrial or heating use. Landfill is a long process. Compaction of waste means that material might only reach a final stable state after about 30 years. Landfill operators are required to have a licence, and to comply with strict waste management regulations. Landfill sites must be designed and constructed to high standards to ensure safe containment and long-term protection.
“Prevention is better than cure”
Another approach to landfill pollution prevention is to minimise domestic waste in the first place. Separation of waste products, composting of organic remains, recycling schemes, and awareness-raising campaigns can all be used by local authorities in the task of domestic waste reduction.
In the Great Stour catchment, 26 sites of waste disposal have been identified by the Environment Agency as potential groundwater contaminants. At present, fortunately, there is no contamination of the river via groundwater leakage of leachate from these sites; they are all are under strict licence, and closely monitored. Landfill remains the most important option locally, the main advantage being the relatively low costs involved. But land shortage, especially in the south-east of England could mean that we will run out of landfill sites within 10 years. Kent County Council has a target that “by 2006, landfill with unprocessed wastes will become the exception in Kent”. The county has the vision of “ultimately aspiring to zero waste”.
Threat from old landfill sites

Land is said to be contaminated when substances are present at concentrations that could be harmful to human life, wildlife or the environment as a whole.
To pose a risk of contamination there must be:
a) A source of contamination b) A pathway of migration c) A specific target
Old landfill provides a ready pollution ‘source’; the slope of the ground and water table generate the ‘pathway’; the nearby river is the pollution ‘target’.
The precise scale and nature of contaminated land in urban areas is often unknown, because there may be few written records of old landfill and other disused ‘brownfield’ industrial sites; local authorities and the Environment Agency will try to compile registers of these as accurately as is possible. Brownfield industrial sites in the Canterbury area, including old waste disposal landfill, provide good opportunities for redevelopment on the outskirts of the city. The Sturry Road park-and-ride car park is a good example. Many of these local sites are close to the River Great Stour, including:
Slow degradation of compostable materials beneath the Sturry Road park-and-ride site is currently releasing ammonia leachate from below. Degradation of old landfill material is a slow process; disposed solids in old waste tips might only reach a final, stable unpolluting state after about 30 years. Meanwhile, discharge of leachate has to be collected in a sump, treated and released slowly into the river under consented agreement with the Environment Agency. Other examples of local brownfield redevelopments (e.g. Safeways, Wincheap) have revealed the presence of contaminated soil, which needs to be carefully removed to avoid pollution of watercourses during the construction phase.

The new microprocessors that Intel Corp. began selling this week will give the company a nine-month sales edge over its closest competitor, Advanced Micro Devices Inc.
The new 45-nanometer chips are made at Intel’s new Fab 32 production facility in Chandler and use technology that can put more transistors on a chip, allowing it to store more data and perform more functions.
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Advanced Micro is not expected to have a product that will compete with it until mid-2008.
Portions of Intel’s new chips also are made with hafnium instead of silicon dioxide, which reduces the electricity leakage encountered as the transistors became smaller. The hafnium reduces power consumption by 30 percent and increases switching speed by 20 percent.
Intel Senior Vice President David Perlmutter called it one of the biggest changes in microprocessor design in the last 40 years
Advanced Micro Devices of Sunnyvale, Calif., is working to upgrade its Dresden, Germany, factory to the 45-nanometer technology, but doesn’t expect to be in full production until the second half of 2008.
Intel shares gained 14 cents, or 0.6 percent, Monday while Advanced Micro fell 42 cents, or 3.4 percent, as most tech stocks stumbled.