General Sinus Principles:  Extension of Trigonometric Functions

General Sinus Principles: Extension of Trigonometric Functions | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article General Sinus Principles: Extension of Trigonometric Functions Mostafa DERRAZ This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9525842/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Classical trigonometric functions $\sin(\alpha)$, $\cos(\alpha)$, and $\tan(\alpha)$ are implicitly defined under an existential condition: the right-angled triangle, or equivalently, vertical projection. This paper presents a rigorous extension of these functions to a bivariate framework by introducing a second variable, the \emph{projection angle}$\phi \in (0,\pi)$, yielding the generalized functions $\sin(\alpha,\phi)$, $\cos(\alpha,\phi)$, and $\tan(\alpha,\phi)$. denotes the studied angle and $\phi$ the angle of projection. The classical functions are recovered as the special case $\alpha + \phi = \pi/2$. A coherent analytical structure is established, encompassing domains, ranges, symmetry identities, reciprocity relations, summation formulas, and a complete differential and integral calculus. In particular, the $n$-th derivative and $n$-th primitive are expressed in closed form via a rotation operator $\omega(z,n)$. (GST), a product-form decomposition of $\sin(\alpha,\phi)$, $\cos(\alpha,\phi)$, and $\tan(\alpha,\phi)$ as infinite products of their scaled counterparts, together with a polynomial approximation analogous to the Taylor--Young expansion of classical trigonometric functions. The geometric interpretation of all six generalized functions on the composed trigonometric circle is developed, and two- and three-dimensional plots illustrate the analytical properties established. A structural analogy between the surface of the general tangent function $\tan(\alpha,\phi)$ and the fundamental mode of the transient heat equation is identified and supported by a numerical fit to real experimental data ($R^2 = 0.83$), suggesting a potential analytical role for the general trigonometric functions in the theory of parabolic partial differential equations. Mathematics Subject Classification (2010). Primary 26A09; Secondary 51N20, 42A05, 26B05, 44A05. generalized trigonometric functions bivariate sine and cosine composed trigonometric circle projection angle General Sinus Transform non-right-angled triangle Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9525842","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":630814926,"identity":"ffbe4bc8-a845-4648-a1f9-4584e8bd1239","order_by":0,"name":"Mostafa DERRAZ","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAyUlEQVRIiWNgGAWjYHACAzDJT7oWyQYgcYAkLQYHiNWi28C88XHlHjt749u9Dx9/qLjDoNvegF+L2QG2YsMzz5KZze4cNzY4cOYZg9kZAnaZHeAxk2w4wMxmdiONTeJg22EGsxsJBLWY/2w4UM9jPAOm5f4DwrYwNhw4LGEgAbcFvw4Gs8NsxUCHHTeQuHOM2eDMmcM8ZmcIOex488aPDQeq7flntzE+qKg4LGd2/AABa5hhDAkIxUNAPTKQIEHtKBgFo2AUjCwAALU4RTcYDaJjAAAAAElFTkSuQmCC","orcid":"","institution":"","correspondingAuthor":true,"prefix":"","firstName":"Mostafa","middleName":"","lastName":"DERRAZ","suffix":""}],"badges":[],"createdAt":"2026-04-25 13:08:13","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9525842/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9525842/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":109209125,"identity":"a3b96cf9-f093-4954-9727-3c75a64cdb9d","added_by":"auto","created_at":"2026-05-13 15:27:34","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":535217,"visible":true,"origin":"","legend":"","description":"","filename":"GeneralSinusPrinciples.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9525842/v1_covered_62d37230-1d45-4d8a-a056-5c222076ddfa.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"General Sinus Principles: Extension of Trigonometric Functions","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"generalized trigonometric functions, bivariate sine and cosine, composed trigonometric circle, projection angle, General Sinus Transform, non-right-angled triangle","lastPublishedDoi":"10.21203/rs.3.rs-9525842/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9525842/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eClassical trigonometric functions $\\sin(\\alpha)$, $\\cos(\\alpha)$, and $\\tan(\\alpha)$ are implicitly defined under an existential condition: the right-angled triangle, or equivalently, vertical projection. This paper presents a rigorous extension of these functions to a bivariate framework by introducing a second variable, the \\emph{projection angle}$\\phi \\in (0,\\pi)$, yielding the generalized functions $\\sin(\\alpha,\\phi)$, $\\cos(\\alpha,\\phi)$, and $\\tan(\\alpha,\\phi)$. denotes the studied angle and $\\phi$ the angle of projection. The classical functions are recovered as the special case $\\alpha + \\phi = \\pi/2$. A coherent analytical structure is established, encompassing domains, ranges, symmetry identities, reciprocity relations, summation formulas, and a complete differential and integral calculus. In particular, the $n$-th derivative and $n$-th primitive are expressed in closed form via a rotation operator $\\omega(z,n)$. (GST), a product-form decomposition of $\\sin(\\alpha,\\phi)$, $\\cos(\\alpha,\\phi)$, and $\\tan(\\alpha,\\phi)$ as infinite products of their scaled counterparts, together with a polynomial approximation analogous to the Taylor--Young expansion of classical trigonometric functions. The geometric interpretation of all six generalized functions on the composed trigonometric circle is developed, and two- and three-dimensional plots illustrate the analytical properties established. A structural analogy between the surface of the general tangent function $\\tan(\\alpha,\\phi)$ and the fundamental mode of the transient heat equation is identified and supported by a numerical fit to real experimental data ($R^2 = 0.83$), suggesting a potential analytical role for the general trigonometric functions in the theory of parabolic partial differential equations.\u003c/p\u003e\n\u003cp\u003eMathematics Subject Classification (2010). 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