[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-571382-105":53,"doc-detail-571382-en":127},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":120,"head_meta":122,"extra_data":124,"updated_unix":126},105,"en","eulers-formula-and-trig-identities-key-formulas-and-properties","Euler's Formula and Trig. Identities - Key Formulas and Properties","","Euler’s formula and trigonometric identities are summarized alongside a set of signal-processing relationships used in continuous-time systems. The document includes key derivations for impulse function properties, convolution and its integral form, and power/energy expressions for signals. It further presents continuous-time Fourier series and Fourier transform statements, Parseval-related relations, and multiple Fourier transform pairs. Additional sections list core Fourier transform properties for operations such as time shifts, scaling, modulation, products, and symmetry for real even/odd signals.",{"@graph":63,"@context":119},[64,80,102],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/eulers-formula-and-trig-identities-key-formulas-and-properties/571382/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/eulers-formula-and-trig-identities-key-formulas-and-properties/571382.png","ImageObject",442,249,{"name":88,"@type":89},"8796093062539","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-24","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":101},"InteractionCounter",{"@type":100},"ViewAction",7,{"@type":103,"mainEntity":104},"FAQPage",[105,111,115],{"name":106,"@type":107,"acceptedAnswer":108},"What is Euler’s formula presented in the document?","Question",{"text":109,"@type":110},"The document states Euler’s formula in complex-exponential form, expressing cosine and sine as real and imaginary parts of e^{jt}. It then uses this to support further trigonometric identities.","Answer",{"name":112,"@type":107,"acceptedAnswer":113},"How does the document define convolution for continuous-time signals?",{"text":114,"@type":110},"It gives the continuous-time convolution integral x(t) * h(t) in the form ∫ x(τ)h(t−τ)dτ over the appropriate limits. This expresses how two time-domain signals combine to form the output.",{"name":116,"@type":107,"acceptedAnswer":117},"Which Fourier-related theorems and properties are included?",{"text":118,"@type":110},"The document includes continuous-time Fourier series and Fourier transform results with Parseval’s theorem, and lists a range of Fourier transform properties covering linearity, time shifting, modulation/frequency shifting, scaling, multiplication/convolution relationships, and symmetry for even/odd real signals.","https://schema.org",{"og:url":78,"og:type":121,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":123,"canonical":78},"index,follow",{"doc_id":125,"site_id":56},571382,1792015748,{"code":4,"msg":5,"data":128},{"doc_id":125,"user_id":129,"nickname":88,"user_avatar":60,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":101,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":9,"language":135,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":61,"update_tm":139,"read_time":4},8796093062539,"Euler's Formula and Trig. Identities ej􀀒 = cos(􀀒) + j sin(􀀒)  \ncos(􀀒) = ~~ ~~12 [ej􀀒 + e􀀀j􀀒 ] sin(􀀒) = ~~ ~~12j [ej􀀒 􀀀 e􀀀j􀀒 ]  \nsin(􀀋 + 􀀌) = sin(􀀋) cos(􀀌) + cos(􀀋) sin(􀀌)  \ncos(􀀋 + 􀀌) = cos(􀀋) cos(􀀌) 􀀀 sin(􀀋) sin(􀀌)  \n1  1  \nsin(􀀋) sin(􀀌) = cos(􀀋 􀀀 􀀌) 􀀀 cos(􀀋 + 􀀌)  \n2 2  \n1  1  \ncos(􀀋) cos(􀀌) = cos(􀀋 􀀀 􀀌) + cos(􀀋 + 􀀌)  \n2 2  \n1  1 sin(􀀋) cos(􀀌) = sin(􀀋 + 􀀌) + sin(􀀋 􀀀 􀀌)  \n2 2 sin2 (􀀋) = ~~ ~~12 (1 􀀀 cos(2􀀋)) cos2 (􀀋) = ~~ ~~12 (1 + cos(2􀀋))  \nImpulse Function Properties g (t)A􀀎 (t 􀀀 t0 ) = g (t0 )A􀀎 (t 􀀀 t0 )  \ng (t0 ) = Z􀀀+11 g (t)􀀎 (t 􀀀 t0 )dt  \nConvolution x (t) 􀀃 h(t) = Z􀀀+11 x (􀀜 )h (t 􀀀 􀀜 )d􀀜  \nPower and Energy Ex = Z􀀀11 jx(t)j 2 dt  \nPx = Tlim!1 1T Z\u003CT > jx(t)j 2 dt  \nContinuous-Time Fourier Series and Parseval's Theorem  \nx (t) =  \nxn =  \nPx =  \n1  \nX xnejn!0 tn= 􀀀1  \n1T0 Z\u003CT0 > x (t)e 􀀀jn!0 tdt 1  \nX jxnj 2  \nn= 􀀀1  \nContinuous-Time Fourier Transform and Parseval's Theorem  \nX (!) = Z􀀀11 x (t)e 􀀀j!tdtx (t) = ~~ ~~21􀀙 Z􀀀11 X (!)ej!td! Ex = ~~ ~~21􀀙 Z􀀀11 jX(!)j 2 d!  \nContinuous-Time  \nej!0 t cos(!0t)  \nsin(!0t) 1  \nx (t) = ( 10;; Tj tj  jTtj sin(!0 t)  \n􀀙t  \n􀀎 (t)  \nu (t)  \n􀀎 (t 􀀀 t0 )  \ne 􀀀atu (t); \u003Cfag > 0 te 􀀀atu (t); \u003Cfag > 0 P+n xnejn!0 t  \nFourier Transform Pairs  \n2􀀙􀀎(! 􀀀 !0 )  \n􀀙 [􀀎(! 􀀀 !0 ) + 􀀎(! + !0 )] j􀀙 [􀀎(! + !0 ) 􀀀 􀀎(! 􀀀 !0 )] 2􀀙􀀎(!)  \n2 sin(!T )  \n!  \nX (!) = ( 10;; !j !0j  j!!0j 1  \nj~~1~~! + 􀀙􀀎(!) e 􀀀j!t0  \n 1   \na+j!  \n 1   \n(a+j!)2  \n2􀀙P+n xn 􀀎 (! 􀀀 n!0 )  \nContinuous-Time Fourier Transform Properties  \nax (t) + by(t) aX (!) + bY(!)  \nx (t 􀀀 t0 ) e 􀀀j!t0 X (!)  \nej!0 tx (t) X (! 􀀀 !0 )  \nx (􀀀t) X (􀀀!)  \nx (at) ~~ ~~j~~ 1~~aj~~ ~~ X 􀀀 ~~!~~​a 􀀁  \nx (t) 􀀃 y(t) X (!)Y (!)  \nx (t)y (t) ~~ ~~2~~1~~􀀙 X (!) 􀀃 Y(!)  \ndxd(tt) j!X (!)  \nR1 x (􀀜 )d􀀜 j~~1~~! X (!) + 􀀙X(0)􀀎(!)  \nx (t) real X (!) = X􀀃 (􀀀!)  \nx (t) real and even X (!) real and even x (t) real and odd X (!) imaginary and odd evenfx(t)g \u003CfX(!)g  \noddfx(t)g j=fX(!)g","cbCainyBTXMqJwWW","https://ap.wps.com/l/cbCainyBTXMqJwWW","pdf",91135,"English","# Euler’s Formula and Trigonometric Identities\n## Impulse Function and Convolution\n## Power and Energy\n## Continuous-Time Fourier Series and Parseval\n## Continuous-Time Fourier Transform and Parseval\n## Fourier Transform Pairs\n## Fourier Transform Properties","[{\"question\":\"What is Euler’s formula presented in the document?\",\"answer\":\"The document states Euler’s formula in complex-exponential form, expressing cosine and sine as real and imaginary parts of e^{jt}. It then uses this to support further trigonometric identities.\"},{\"question\":\"How does the document define convolution for continuous-time signals?\",\"answer\":\"It gives the continuous-time convolution integral x(t) * h(t) in the form ∫ x(τ)h(t−τ)dτ over the appropriate limits. This expresses how two time-domain signals combine to form the output.\"},{\"question\":\"Which Fourier-related theorems and properties are included?\",\"answer\":\"The document includes continuous-time Fourier series and Fourier transform results with Parseval’s theorem, and lists a range of Fourier transform properties covering linearity, time shifting, modulation/frequency shifting, scaling, multiplication/convolution relationships, and symmetry for even/odd real signals.\"}]","Euler's Formula and Trig. Identities - Key Formulas and Properties | PDF",1791914555]