Alg1131W8T2 - Powers of sine and cosine

Saturday, 21 April 2018

11:24 AM

Machine generated alternative text: The similarity between De Moivre's theorem (cos t + i sin and the relation is a partial motivation for introducing the definition it e It then follows that De Moivre's relation can be rewritten as cos nt + i sin nt nt cos t + i sin t. int e This is a remarkable and powerful connection between the circular functions cost and sin t, and the exponential function e probably never guess if we restricted ourselves to real number arithmetic. Replacing t with —t gives the relation which we would —it e a cos t + b sin t, where a = 1 Note: the Maple notation for the complex unit i is I (captial i). —it If we add and subtract the formulas for ect and e we get and sin t —it cos t 2 (exp(lt) - exp(-lt))/(21) Note: the complex exponential ect is written in Maple notation as exp (I* t) . These are very important formulas. In particular, they allow us to get interesting expressions for the powers of cost and sin t.

 

Machine generated alternative text: Let's express cos t in terms of cosines and sines of multiples of t. and the expression For this we need recall the Binomial theorem for n = —it 5, namely that We group together like terms and use the facts that cos t —int 2 —int and to get that where a = 1/16 cos nt 5/16 e sin nt 2 cos t a cos 5t + b cos 3t + c cos t 10/16

 

Machine generated alternative text: Let's express sin t in terms of cosines and sines of multiples oft. For this we need the Binomial theorem for n = 4, it —it sin t Don't forget that important i in the denominator! We must group together like terms and use the facts that int along with the expression —int to get that where a 118 o b -1/2 O cos nt and c — int t and sin nt sm 6/16 a cos4t + bcos2t + c

 

Machine generated alternative text: Here is a graph of the function y = cos x, along with the linear combination acosx + bcos3x + ccos 5m. Your job is to fiddle with the sliders to adjust the values of a; b and c to make the linear combination match up with y engineers and physicists, going back to JL. Fourier in 1830 and his study of the heat equation. a = 0.63 b=032 cos x. This is basic Fourier analysis; a very important topic for —3 c = 0.07 -2 2 My best values were a

 

Machine generated alternative text: This kind of decomposition of a wave pattern into frequencies is bread and butter for electrical engineers' This is why they are all very keen on complex numbers and their magic. Here is a similar question as the last; but this time you want to create the wave y = sin x using a linear combination of the constituent functions 1, cos2x and cos4x to form a + b cos2x + c cos4x. How much of each do you need? a = 0.3 2 b = -0 49 c: 0.15 -2 My best values were a 2 37 3 .15

 

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