Geometry and Trigonometry

Geometry and Trigonometry — page 3 — Geometry and Trigonometry that shooting sonar waves from one focus makes sounds that bounce off the inside walls of the ellipsoid and they all strike the other focus at once like a gentle hammer That's how doctors use those sounds special instruments such as a lithotripter machine and ellipsoids to help break up kidney stones in the patient No surgery is needed The patient makes a faster recovery than if surgery had been used His or her kid ney stones are painlessly shattered with no harm to the kidney and the patient might go to school or work the very next day When readers enter the pages of this book and see the basics of geometry and trigonom etry it pays to ask themselves with each figure they see could they discover some great inven tion with this shape Could they save lives Could they ease pain Could they somehow use it to create a new invention What prob lem might they solve with it Students will discover why trigonometry is called that and they ll see a math topic begin to unfold that strangely they have already used quite a bit even though they might not have been aware of it Trigonometry is embedded in the software of computers When people have played a computer game where their car for instance spins and misses trouble on the road the joystick is really telling the computer

Geometry and Trigonometry — page 4 — INTRODUCTION to use trigonometry formulas to get the car to do what the players want So the math comes alive for these individuals They'll encounter in this book some basics of trigonometry often fondly called trig Finally readers will also find fascinating information on great mathematicians who made the ideas of geometry and trigonome try apply to wonderful discoveries that have changed the world they live

Geometry and Trigonometry — page 5 — EUCLIDEAN GEOMETRY he ancient branch of mathematics known as geometry deals with points lines sur faces and solids and their relationships In particular geometry may be thought of as offering W precise definitions of many dif ferent figures 2 construction methods for drawing figures 3 a wealth of facts about the figures and most important 4 ways to prove the facts Geometry was thoroughly organized in about 300 BCE when the Greek mathemati cian Euclid gathered what was known at the time added original work of his own and arranged 465 propositions into 13 books col lectively called Elements The books covered not only plane and solid geometry but also much of what is now known as algebra trigo nometry and advanced arithmetic 10

Geometry and Trigonometry — page 6 — EUCLIDEAN GEOMETRY Down through the ages the propositions have been rearranged and many of the proofs are different but the basic idea presented in the Elements has not changed In the work facts are not just cataloged but are developed in an orderly way starting with statements defini tions common notions and postulates that seem perfectly self evident with each succes sive theorem proved by using only previously shown facts This mode of reasoning known as the axiomatic method has profoundly influ enced epistemology the study of the nature origin and limits of human knowledge and education Even in 300 BCE geometry was recognized to be not just for mathematicians Anyone can benefit from the basic teachings of geometry which are how to follow lines of reasoning how to say precisely what is intended and especially how to prove basic concepts by fol lowing these lines of reasoning Taking a course in geometry is beneficial for all students who will find that learning to reason and prove convincingly is necessary for every profession It is true that not everyone must prove things but everyone is exposed to proof Politicians advertisers and many other people try to offer convincing arguments Anyone who cannot tell a good proof from a bad one may easily be persuaded in the wrong direction Geometry 11

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