Pythagoras, Trigonometry, and The Cell Phone

Transcription

1 Presentation to Clark State Communit College 03 Ma 2006 Pthagoras, Trigonometr, and The Cell Phone B 2 C 2 A 2 C 2 =A 2 +B 2

2 Poem: Love Triangle Consider ol Pthagorus, A Greek of long ago, And all that he did give to us, Three sides whose squares now show In houses, fields, and highwas straight; In buildings standing tall; In might planes that leave the gate; And, micro-sstems small. Yes, all because he got it right When angles equal ninet One geek (BC), his plane delight One world changed aplent!

3 An Earl Pre-Algebraic Visual Proof of the Pthagorean Theorem

4 Trigonometr is Pthagorean Right-Triangle Geometr P ( t) = ( x, ) C ( x, ) r =1 α ( 0,0) x (1,0 ) t A α B 1 α ( 0,0) x (1,0 ) sin(α ) = sin( t ) = cos(α ) = cos( t ) = x sin(α ) = = 1 BC AC x cos(α) = = 1 AB AC Dnamic view of trigonometr as a descriptive measurement sstem for right-triangular relationships that propagate through time. Static view of trigonometr as a descriptive measurement sstem for right-triangular relationships that occur in phsical space. In either case, since these relationships sin, cosine, etc are derived using right triangles, we can sa all trigonometr is Pthagorean in origin.

5 Wh the Human Voice is Trigonometric Friendl (One) r =1 P ( t) = ( x, ) t sin(α) = sin( t ) = α ( 0,0) x (1,0 ) sin(α ) = sin( t ) = cos(α ) = cos( t ) = x A basic periodic sinusoidal waveform Same waveform accelerated Called a frequenc change Same waveform amplified

6 Wh the Human Voice is Trigonometric Friendl (Two) Amplified plus frequenc change Amplified plus frequenc change plus sign change to alwas positive: rectification in electrical engineering Effect of adding or subtracting two or more rectified sine and cosine waves.

7 Wh the Human Voice is Trigonometric Friendl (Three) Hi Jim Sound wave Hi Jim Sound wave converted to electromagnetic long waves using cellular technologies that produce electronic trigonometric equivalents Hi Jim Cell Tower Power Boost John, ou sound a little frogg toda. Got a cold?

8 Meet the Cell in Cellular Phone Yak, ak Standard Seven-Cell Cluster 832 federall-assigned frequencies 48 channels reserved for network control 784 channels for voice communications Cell-Phone conversations are simultaneous and two-wa Each conversation requires two different electromagnetic frequencies Each frequenc is a specific waveform patterned after a basic additive combination of sines and cosines Mathematicall processed per the rules governing sines and cosines using electronic equipment designed to do just that Our basic seven-cell cluster can accommodate 392 separate communications 56 simultaneous communications per cell Yak, ak, goodbe The beaut of the cell sstem is that 832 frequencies can be reused throughout the provider s assigned area, since a cellular sstem b design is a low-power sstem

9 Some Geometricall Unacceptable Cellular Clusters 4-Cell Cluster 6-Cell Cluster 5-Cell Cluster

10 Trigonometric Geometr of a Tpical Single Cell All Cellular Infrastructure has to be precisel located per surveing techniques, which are based on trigonometr 2 miles miles 4.0 miles Trigonometric Waveforms! Here, Our Cell Area =12x(1/2)(1.155)(2)=13.85 square miles Even the height of that cell tower has to be precisel set per the earth s curvature so none of its emitted signals interfere with those of the six surrounding towers. Again, trigonometr is the mathematical tool required to do this.

11 Surveing Cellular Tower Placement Initial placement done b use of GPS sstem Heavil dependent on trigonometric principles Placement fine-tuned b traditional surveing Traditional surveing is founded on two major tools, which allow trigonometr to expand capabilit from right triangles to general triangles Law of Cosines Law of Sines C A β a x γ h x + = c b α B a 2 = c 2 + b 2 2bccos( α) Law of Cosines b a c = = sin( β ) sin( α) sin( γ ) Law of Sines

12 Cell Phones, Cell Towers, and Cellular Technolog as a Dnamic Snthesis of Trigonometric Principles.

13 Summar Technologicall speaking, the totalit of the cellular industr rests on trigonometric principles Throughout this course, we will return to the cellular industr again and again as we learn various aspects of the trigonometric language and discipline Will work applications directl related to corresponding technical aspects of the cellular industr Will also discuss the various jobs available in the booming worldwide cellular industr, man of which require the technical skills that ou will be learning in this course! Pthagoras, if ou could onl hear us now! Thank ou!