{"id":10,"date":"2007-11-08T15:10:13","date_gmt":"2007-11-08T14:10:13","guid":{"rendered":"http:\/\/www.walkingrandomly.com\/?p=10"},"modified":"2010-01-30T21:28:35","modified_gmt":"2010-01-30T20:28:35","slug":"my-favourite-formula","status":"publish","type":"post","link":"https:\/\/walkingrandomly.com\/?p=10","title":{"rendered":"My favourite formula"},"content":{"rendered":"

Are you geeky enough to have a favourite formula? Because I am and it turns out that I am not the only one either. I have just googled ‘favourite formula’ and the second result was a link to the aply named ‘Mathematics Weblog<\/a>‘ where the author described an equation usually known as Euler’s formula.<\/p>\n

\" \light e^{i \pi} = -1\"<\/p>\n

The Nobel prize winning physicist, Richard Feynman, once referred to this equation as ‘the most remarkable formula in mathematics’ <\/em>and he really knew his stuff. Now I agree with Feynman, it really is an amazing formula since it connects some of the most important constants of mathematics in a small and elegant package, but it’s not my favourite. Elegant it may be but it’s not particularly useful.<\/p>\n

My personal favourite is the general case of the above equation. Also called Euler’s equation, it is written as<\/p>\n

\" \light e^{ix}=\cos(x)+ i \sin(x)\"<\/p>\n

Set \" \light x\" to \"\light \pi\" and out pops Feynman’s favourite but we can do so much more with it than that and in this post I am going to focus on using it to obtain the trigonometric addition identities.<\/p>\n

<\/p>\n

I have always struggled to remember trig identities – No matter how hard I try, I somehow manage to get the signs wrong but with a little bit of work I can just derive them using Euler’s equation and be sure that I have got them right. Start off by writing<\/p>\n

\" \light e^{i(a+b)} = e^{ia}e^{ib}\"<\/p>\n

now substitute Euler’s formula in both sides of the equation<\/p>\n

\" \light cos(a+b) + i sin(a+b) = (cos(a) + i sin(a)) (cos(b)+i sin(b))\"<\/p>\n

Multiplying out the right hand side leads to<\/p>\n

\" \light cos(a+b) + i sin(a+b) = cos(a) cos(b)-sin(a) sin(b) + i ( cos(a)sin(b) + sin(a)cos(b) ) \"<\/p>\n

Equating real and imaginary parts on both sides gives us two trig identities for the price of one<\/p>\n

\" \light cos(a+b) = cos(a) cos(b)-sin(a) sin(b) \"<\/p>\n

\" \light sin(a+b) = cos(a)sin(b) + sin(a)cos(b) \"<\/p>\n

We can get similar equations for \" \light cos(a-b) \" and \" \light sin(a-b) \" by simply substituting -b for b.<\/p>\n

In my dim and distant past I used to be an assistant instructor for undergraduate physics problems classes and I found that many students had not seen this derivation before but they felt that it was obvious (and rather useful) once they had been shown it.<\/p>\n","protected":false},"excerpt":{"rendered":"

Are you geeky enough to have a favourite formula? Because I am and it turns out that I am not the only one either. I have just googled ‘favourite formula’ and the second result was a link to the aply named ‘Mathematics Weblog‘ where the author described an equation usually known as Euler’s formula. The […]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[6],"tags":[],"class_list":["post-10","post","type-post","status-publish","format-standard","hentry","category-general-math"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p3swhs-a","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/posts\/10","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=10"}],"version-history":[{"count":5,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/posts\/10\/revisions"}],"predecessor-version":[{"id":2250,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/posts\/10\/revisions\/2250"}],"wp:attachment":[{"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=10"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=10"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=10"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}