MHB Trigonometric of tangent and sine functions

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The discussion focuses on simplifying the expression involving tangent and sine functions: \(\left(\tan \dfrac{2\pi}{7}-4\sin \dfrac{\pi}{7}\right)\left(\tan \dfrac{3\pi}{7}-4\sin \dfrac{2\pi}{7}\right)\left(\tan \dfrac{6\pi}{7}-4\sin \dfrac{3\pi}{7}\right). A user named Dan shares a numerical approximation of the result, approximately -8.7083, and asks if they are close to the correct answer. The conversation revolves around verifying this simplification and discussing the properties of the trigonometric functions involved. The overall goal is to confirm the accuracy of the numerical result and the simplification process.
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Simplify $\left(\tan \dfrac{2\pi}{7}-4\sin \dfrac{\pi}{7}\right)\left(\tan \dfrac{3\pi}{7}-4\sin \dfrac{2\pi}{7}\right)\left(\tan \dfrac{6\pi}{7}-4\sin \dfrac{3\pi}{7}\right)$.
 
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I got about -8.7083121069873265814843145114959219999438416220762493062309701968483080116471438626714199913273209202440106287205185869613863265722279384849632901398226721432702744787955199077781973792055631357781755415456733331388124469116498805756011983384506433640247288593850820089590928253650738597742412754348439402301589178427587352708579714081417809559865987679591010005488147679340458470850085945675037017144904042287279335497160556553452963874716319026939725908000014025345344112227718546669183167834689374961602264946845122520973969612465747159538537772604016219227651262298967156522926816165972425283143645081705710618441263404832174787619393871431981552587401078586609898568054592224581007986448669288148444215128498186229896741206349315952054866256984612864167199842472866575465952447085845790874114020207148838309575537421068097653460699858226519033408205869208586639796284569971790165922274544500088737294024496093237634962135759638417523292668843921768916664652250085254395250841082981914897137469833874653795905403859395226103590166597307639852476273392879083873723649605334228340639098097814440444549718831560592044450598723461692804007733588937741590648889021935199757480637837660523461803937847078950866107512314217003731198786042996011750451712135884514444878781114357178952544416819768594073353584110967734337432198244086867356343864416552649586451520413569547092694404444532795994994438271445333716459116502792251091594744077024600155160457563503684373664211499889132746530159707148096441069149624420565803877227318131627873671715105932608979648146278844613590157353436714676787933231516958800501704045205913556958442328349254532043183475963879427381686337157339601066369199608533673687386762949721994907024739645071023865884051877078161059364899178505753714909479228031574944638783941268199880771092085126999780242627307210910698666574484989827761671955524479251591513340499083326461726929118742316319408632872884749801756534634165971359295929387392681708974873301182019170001880197471302122860499952496977...

Am I close?

-Dan
 
Thread 'Erroneously finding discrepancy in transpose rule'

Thread 'Erroneously finding discrepancy in transpose rule'

Obviously, there is something elementary I am missing here. To form the transpose of a matrix, one exchanges rows and columns, so the transpose of a scalar, considered as (or isomorphic to) a one-entry matrix, should stay the same, including if the scalar is a complex number. On the other hand, in the isomorphism between the complex plane and the real plane, a complex number a+bi corresponds to a matrix in the real plane; taking the transpose we get which then corresponds to a-bi...
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