Angličtina pro matematiky IV

COURSE MATERIALS AND HOMEWORK WEEK VI.

 

TRIGONOMETRIC FUNCTIONS II
(materials adapted from Mathematics, encyclopedia articles on http//en.citizendium.org)
1. What are the methods of calculating trigonometric functions? Try to list some.
    .....................................................................   
    ....................................................................
   Now read the short text and try to check your answers and add any knew
information.
Calculating trigonometric functions
Trigonometric functions were among the earliest uses for mathematical tables. Such tables were incorporated into mathematics textbooks and students were taught to look up values and how to interpolate between the values listed to get higher accuracy. Slide rules had special scales for trigonometric functions.
Today scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan and sometimes cis) and their inverses. Most allow a choice of angle measurement methods: degrees, radians and, sometimes, Grad. Most computer programming languages provide function libraries that include the trigonometric functions. The floating point unit hardware incorporated into the microprocessor chips used in most personal computers have built-in instructions for calculating trigonometric functions.
3. There are some definitions of the terms from the article. Try to find the terms.
A. a part of a computer system specially designed to carry out operations on floating point numbers.
B. a system for numerical representation in which a string of digits (or bits) represents a rational number.
C. a unit of plane angle, equivalent to 1⁄400 of a full circle, dividing a right angle in 100. 
D. a type of electronic calculator, usually but not always handheld, designed to calculate problems in science (especially physics), engineering, and mathematics.
4. Try to replace words in italics with synonyms.
Notation
5. Study the following information, then have a look at the table and try to fill in the missing parts.
Angles
This article uses Greek letters such as alpha (α), beta (β), gamma (γ), and theta (θ) to represent angles. Several different units of angle measure are widely used, including degrees, radians, and grads:
1 full circle  =  360 degrees  =  2π radians  =  400 grads.
The following table shows the conversions for some common angles:

30°
60°
120°
150°
 
240°
300°
330°
 
33⅓ grad
66⅔ grad
 
166⅔ grad
233⅓ grad
266⅔ grad
 
366⅔ grad
 
45°
90°
 
180°
225°
270°
315°
 
 
 
50 grad
 
150 grad
200 grad
 
300 grad
350grad
400 grad

Unless otherwise specified, all angles in this article are assumed to be in radians, though angles ending in a degree symbol (°) are in degrees.
6. Read the following part and fill in the missing information.
Trigonometric functions
The primary trigonometric functions are the sine and cosine of an angle. These are usually abbreviated sin(θ) and cos(θ), respectively, where θ is the a................. In addition, the p……….. around the angle are sometimes omitted, e.g. sin θ and cos θ.
The tangent (tan) of an angle is the r ......................of the sine to the cosine:
Finally, the reciprocal functions secant (sec), cosecant (csc), and c………… (cot) are the r.......................of the cosine, sine, and tangent:
These definitions are sometimes  r…………….. to as ratio identities.
The Pythagorean identity
The basic relationship between the sine and the cosine is the Pythagorean trigonometric identity:
This can be viewed as a version of the P............................., and follows from the equation x2 + y2 = 1 for the unit circle. This e.....................can be solved for either the sine or the cosine:
Related identities
Dividing the Pythagorean identity through by either cos2 θ or sin2 θ yields two other i................................:
Using these identities together with the ratio identities, it is possible to .................. any trigonometric function in terms of any other (up to a plus or minus sign).
 
                 Solve these problems:
1.      The angle of elevation of the Rock Mountain fire-control tower from the top of Blue Mountain 3 miles away (horizontal distance) is 18°. How much higher than Blue Mountain is the fire-control tower?
2.      A pendulum 40 inches long is moved 30° from the vertical. How much is the lower end of the pendulum lifted?
3.      Find the lenght of the altitude of an isosceles triangle whose base has lenght 20 inches and whose base angles each has measure 45°.
      
 
 
HW: Ask Qs so that you can answer with the underlined word(s).
1)      ……………………………………………………………………
2)      ……………………………………………………………………
3)      …………………………………………………………………….
4)      …………………………………………………………………….
5)      …………………………………………………………………….
6)      …………………………………………………………………….

 

 

Great maths puzzle 'solved'
By Dr David Whitehouse
BBC News Online science editor
 
1) Explain what is: a) Poincare Conjecture
                        b) Fermat´s last theorem

 
2) Replace the underlined words with their synonyms.  
A Russian mathematician claims to have proved the Poincare Conjecture, one of the most famous problems in mathematics.
Dr Grigori Perelman, of the Steklov Institute of Mathematics of the Russian Academy of Sciences, St Petersburg, has been touring US universities describing his work in a series of papers not yet completed.
The Poincare Conjecture, an idea about three-dimensional objects, has haunted mathematicians for nearly a century. If it has been solved, the consequences will reverberate throughout geometry and physics.
If his proof is accepted and survives two years of scrutiny, Perelman could also be eligible for a $1m prize sponsored by the Clay Mathematics Institute in Massachusetts for solving what the centre describes as one of the seven most important unsolved mathematics problems of the millennium.
Spheres and doughnuts
Formulated by the remarkable French mathematician Henri Poincare in 1904, the conjecture is a central question in topology, the study of the geometrical properties of objects that do not change when they are stretched, distorted or shrunk.
For example, the hollow shell of the surface of the Earth is what topologists call a two-dimensional sphere. It has the property that every lasso of string encircling it can be pulled tight to a point.
On the surface of a doughnut however, a lasso passing through the hole in the centre cannot be shrunk to a point without cutting through the surface meaning that, topologically speaking, spheres and doughnuts are different.
Since the 19th Century, mathematicians have known that the sphere is the only enclosed two-dimensional space with this property. But they were uncertain about objects with more dimensions.
The Poincare Conjecture says that a three-dimensional sphere is the only enclosed three-dimensional space with no holes. But the proof of the conjecture has eluded mathematicians.
Poincare himself demonstrated that his earliest version of his conjecture was wrong. Since then, dozens of mathematicians have asserted that they had proofs until fatal flaws were found.
Internet rumours
Rumours about Perelman's work have been circulating since November, when he posted the first of his papers reporting the result on an internet preprint server.
Since then, Perelman has persistently declined to be interviewed, saying any publicity would be premature.
Dr Tomasz Mrowka, a mathematician at the Massachusetts Institute of Technology, said: "It's not certain, but we're taking it very seriously.
"We're desperately trying to understand what he has done here," he adds.
Some are comparing Perelman's work with that of Andrew Wiles, who famously solved Fermat's Last Theorem a decade ago.
Indeed, Wiles was in the Taplin Auditorium at Princeton University, New Jersey, where he holds a chair in mathematics, to hear Perelman describe his work recently. Behind him sat John Nash, the Nobel Laureate who inspired the film A Beautiful Mind.
Million dollar afterthought
What is all the more remarkable about Perelman's proposal is that he is trying to achieve something far grander than merely solving Poincare's Conjecture.
He is trying to prove the Geometrisation Conjecture proposed by the American mathematician William Thurston in the 1970s - a far more ambitious proposal that defines and characterises all three-dimensional surfaces.
"He's not facing Poincare directly, he's just trying to do this grander scheme," said Professor Peter Sarnak, of Princeton.
After creating so much new mathematics, the Poincare result is just "a million dollar afterthought," he said.
If Perelman has solved Thurston's problem then experts say it would be possible to produce a catalogue of all possible three-dimensional shapes in the Universe, meaning that we could ultimately describe the actual shape of the cosmos itself.
Story from BBC NEWS:
http://news.bbc.co.uk/go/pr/fr/-/2/hi/science/nature/3005875.stm

Published: 2003/05/07 12:44:52 GMT


3) Find out who these scholars are:
    a) Andrew Wiles
    b) William Thurston
    c) Tomasz Mrowka
    d) Peter Sarnak
    e) John Nash
    f) Grigori Perelman
 

 

Listening 5
http://ocw.mit.edu/OcwWeb/Mathematics/18-02Fall-2007/VideoLectures/detail/embed02.htm
Answer the Qs.
  1. What was the topic of the last lecture?
  2. How is the dot product obtained?
  3. How do you get the scalar?
  4. What is the geometric interpretation of that?
  5. What are the applications of that?
a)   To find angles and ………………………
b)   To find out whether the vectors are……………
c)    To find components of a vector A ………………
  1. What is the dot product between A and u?
  2. What is 100?
  3. Where can you use the Newton´ s Laws of Mechanics?
  4. What example of that use does he give?
 
 
 
 
 
 
 
 
Listen again and try to fill in the missing sentences.
                                Well, how do we find that?
That is very easy.
                                                       So what is an application of that?
                             Let's say that ……..
                                 Let me explain.                           For example
We have seen several applications of that.
 
 
 
Transcript - Lecture 2
Thank you. Let's continue with vectors and operations of them. Remember we saw the topic yesterday was dot product. And remember the definition of dot product, well, the dot product of two vectors is obtained by multiplying the first component with the first component, the second with the second and so on and summing these and you get the scalar.
And the geometric interpretation of that is that you can also take the length of A, take the length of B multiply them and multiply that by the cosine of the angle between the two vectors. One application is to find lengths and angles. For example, you can use this relation to give you the cosine of the angle between two vectors is the dot product divided by the product of the lengths.
Another application that we have is to detect whether two vectors are perpendicular. To decide if two vectors are perpendicular to each other, all we have to do is compute our dot product and see if we get zero. And one further application that we did not have time to discuss yesterday that I will mention very quickly is to find components of, let's say, a vector A along a direction u. So some unit vector. Let´s say that I have some direction. The horizontal axis on this blackboard. But it could be any direction in space. And, to describe this direction, maybe I have a unit vector along this axis. I have any of a vector A and I want to find out what is the component of A along u.
That means what is the length of this projection of A to the given direction? This thing here is the component of A along u. Well, we know that here we have a right angle. So this component is just length A times cosine of the angle between A and u. But now that means I should be able to compute it very easily because that's the same as length A times length u times cosine theta because u is a unit vector. It is a unit vector.
That means this is equal to one. And so that's the same as the dot product between A and u. And, of course, the most of just cases of that is say, for example, we want just to find the component along i hat, the unit vector along the x axis. Then you do the dot product with i hat, which is 100. What you get is the first component. And that is, indeed, the x component of a vector.
Similarly, say you want the z component you do the dot product with k that gives you the last component of your vector. But the same works with a unit vector in any direction. Well, for example, in physics maybe you have seen situations where you have a pendulum that swings. You have maybe some mass at the end of the string and that mass swings back and forth on a circle. And to analyze this mechanically you want to use, of course, Newton's Laws of Mechanics and you want to use forces and so on, but I claim that components of vectors are useful here to understand what happens geometrically.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Transcript - Lecture 2
Thank you. Let's continue with vectors and operations of them. Remember we saw the topic yesterday was dot product. And remember the definition of dot product, well, the dot product of two vectors is obtained by multiplying the first component with the first component, the second with the second and so on and summing these and you get the scalar.
And the geometric interpretation of that is that you can also take the length of A, take the length of B multiply them and multiply that by the cosine of the angle between the two vectors. We have seen several applications of that. One application is to find lengths and angles. For example, you can use this relation to give you the cosine of the angle between two vectors is the dot product divided by the product of the lengths.
Another application that we have is to detect whether two vectors are perpendicular. To decide if two vectors are perpendicular to each other, all we have to do is compute our dot product and see if we get zero. And one further application that we did not have time to discuss yesterday that I will mention very quickly is to find components of, let's say, a vector A along a direction u. So some unit vector. Let me explain. Let's say that I have some direction. For example, the horizontal axis on this blackboard. But it could be any direction in space. And, to describe this direction, maybe I have a unit vector along this axis. Let's say that I have any of a vector A and I want to find out what is the component of A along u.
That means what is the length of this projection of A to the given direction? This thing here is the component of A along u. Well, how do we find that? Well, we know that here we have a right angle. So this component is just length A times cosine of the angle between A and u. But now that means I should be able to compute it very easily because that's the same as length A times length u times cosine theta because u is a unit vector. It is a unit vector.
That means this is equal to one. And so that's the same as the dot product between A and u. That is very easy. And, of course, the most of just cases of that is say, for example, we want just to find the component along i hat, the unit vector along the x axis. Then you do the dot product with i hat, which is 100. What you get is the first component. And that is, indeed, the x component of a vector.
Similarly, say you want the z component you do the dot product with k that gives you the last component of your vector. But the same works with a unit vector in any direction. So what is an application of that? Well, for example, in physics maybe you have seen situations where you have a pendulum that swings. You have maybe some mass at the end of the string and that mass swings back and forth on a circle. And to analyze this mechanically you want to use, of course, Newton's Laws of Mechanics and you want to use forces and so on, but I claim that components of vectors are useful here to understand what happens geometrically.
 
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