Class Note for MATH 111 at UA
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This 2 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 16 views.
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Date Created: 02/06/15
Derivation of sum and difference identities for sine and cosine John Kerl March 26 2007 The authors of your trigonometry textbook give a geometric derivation of the sum and difference identities for sine and cosine I nd this argument unwieldy 7 I don7t expect you to remember it in fact I don7t remember it There7s a standard algebraic derivation which is far simpler However to use it you need complex arithmetic which we don7t cover in Math 111 Nonetheless I will present the derivation so that you will have seen how simple the truth can be and so that you may come to understand it after you7ve had a few more math courses And in fact all you need are the following facts 0 Complem numbers are of the form ab where a and b are real numbers andz is de ned to be a square root of 71 That is 2392 71 Of course 702 71 as well so 7239 is the other square root of 71 o The number a is called the real part of a big the number b is called the imaginary part of a In All the real numbers you7re used to working with are already complex numbers 7 they simply have zero imaginary part 0 To add or subtract complex numbers add the corresponding real and imaginary parts For example 2 3239 plus 4 5239 is 6 8239 0 To multiply two complex numbers a In and c dz39 just FOIL out the product a In c di and use the fact that 2392 71 Then collect like terms 0 The familiar exponential function f em takes real valued input However it can be extended to take complex valued input All the usual rules for exponents apply so 6abl 6046171 We compute e as always 7 this is the same exponential function as always The question is what does it mean to raise e to an imaginary power I assert to you that we write 6M cosb z sinb 1 where the cosine and sine functions are as usual This famous formula is called Euler s formula Euler is pronounced Oiler You can read all about this formula on Wikipedia 7 also see their nice article on the complex numbers Given these facts7 we can simply write down what Ma a is the sum and difference formulas for sine and cosine fall out as a consequence Using the usual rules for exponents7 we can write this as 6iuz eiaei 39 Now all we need to do is write out the two sides using Euler7s formula The left hand side is 6M cosa B z sinoz 6 Using the de nition7 FOlLing7 and collecting like terms7 the right hand side is ewe cos 04 239 sin 04 cos B z sin B cos Oz cos 7 sina sin 6 z sinoz cos B cos Oz sin 6 Equating real and imaginary parts of the left hand side and the right hand side gives us7 two for the price of one7 the familiar sum identities for sine and cosine sina B sin 04 cos B cos Oz sin cosoz cosozcos isinozsin Repeat this for ei to get the difference identities You can do that 7 just remember that cosine and sine are even and odd functions7 respectively7 so cos7 cos and sin7 isin ln summary7 we have sina i B cosa i B cos Oz cos sin 04 sin 6 sina cos i cos asin V
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