Why are the trig functions versine, haversine, exsecant, etc, seldom utilized in present society?

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Heretofore, I was browsing through a Wikipedia article about the trigonometric identities, when I came across something quite peculiar, namely forgotten trigonometric functions.



The versine (arguably the most basic of the functions), coversine, haversine and exosecant formulas had once been utilised for navigational purposes, prior to GPS tracking systems. However, recently, they have become less common in mathematical society and beyond. Why is that?



Here is a link to a PDF file describing all of these now-obsolete trig functions: "The Forgotten Trigonometric Functions, or
How Trigonometry was used in the Ancient Art of Navigation (Before GPS!)"










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    $begingroup$


    Heretofore, I was browsing through a Wikipedia article about the trigonometric identities, when I came across something quite peculiar, namely forgotten trigonometric functions.



    The versine (arguably the most basic of the functions), coversine, haversine and exosecant formulas had once been utilised for navigational purposes, prior to GPS tracking systems. However, recently, they have become less common in mathematical society and beyond. Why is that?



    Here is a link to a PDF file describing all of these now-obsolete trig functions: "The Forgotten Trigonometric Functions, or
    How Trigonometry was used in the Ancient Art of Navigation (Before GPS!)"










    share|cite|improve this question









    New contributor




    Quantum Entanglement is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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      $begingroup$


      Heretofore, I was browsing through a Wikipedia article about the trigonometric identities, when I came across something quite peculiar, namely forgotten trigonometric functions.



      The versine (arguably the most basic of the functions), coversine, haversine and exosecant formulas had once been utilised for navigational purposes, prior to GPS tracking systems. However, recently, they have become less common in mathematical society and beyond. Why is that?



      Here is a link to a PDF file describing all of these now-obsolete trig functions: "The Forgotten Trigonometric Functions, or
      How Trigonometry was used in the Ancient Art of Navigation (Before GPS!)"










      share|cite|improve this question









      New contributor




      Quantum Entanglement is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.







      $endgroup$




      Heretofore, I was browsing through a Wikipedia article about the trigonometric identities, when I came across something quite peculiar, namely forgotten trigonometric functions.



      The versine (arguably the most basic of the functions), coversine, haversine and exosecant formulas had once been utilised for navigational purposes, prior to GPS tracking systems. However, recently, they have become less common in mathematical society and beyond. Why is that?



      Here is a link to a PDF file describing all of these now-obsolete trig functions: "The Forgotten Trigonometric Functions, or
      How Trigonometry was used in the Ancient Art of Navigation (Before GPS!)"







      trigonometry math-history spherical-trigonometry






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      Quantum Entanglement is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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      edited 30 mins ago









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          $begingroup$

          Those functions are much less used than before for one reason: the advent of electronic computers.



          Before that, one had to rely either on tables or on slide rules. Tables were usually table of logarithms, and they included the logarithms of trigonometric functions as well. The trigonometric functions were then useful not only for geometric applications, but also to simplify algebraic calculations with logarithms.



          For instance, to compute $logsqrt{a^2+b^2}$ when $log a$ and $log b$ are known, you could find $theta$ such that $logtantheta=logfrac ba=log b-log a$, then $logsqrt{a^2+b^2}=log a+logsqrt{1+tan^2theta}$ and $logsqrt{1+tan^2theta}=logfrac{1}{cos theta}=-logcostheta$. There are many similar formulas.



          For geometric applications, sometimes versine and similar functions allow computing with greater precision while not adding too much computation. See for instance the haversine formula used to compute great circle distance (useful in navigation). The straightforward formula with arccosine has poor accuracy when the angle is small (the most common case), due to the fact that cosine is flat at $0$. However the formula with haversine is more accurate. To achieve the same, you would have to use $sin^2(theta/2)$ everywhere, which require more computations (but it's still reasonable with logarithms). Therefore, navigation table like Nories's nautical tables have an haversine table.



          Note that tables of logarithms are usually accurate up to 5 digits (some larger tables had 7 digits, some very special ones had better precision but were difficult to use : more digits = much more space on paper). Slide rules have roughly 3 digits of precision.



          All of this is rendered pretty useless with calculators, which have usually around 15 digits of precision and compute fast enough that we don't have to worry about speeding things up with extra functions.






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            1 Answer
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            active

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            6












            $begingroup$

            Those functions are much less used than before for one reason: the advent of electronic computers.



            Before that, one had to rely either on tables or on slide rules. Tables were usually table of logarithms, and they included the logarithms of trigonometric functions as well. The trigonometric functions were then useful not only for geometric applications, but also to simplify algebraic calculations with logarithms.



            For instance, to compute $logsqrt{a^2+b^2}$ when $log a$ and $log b$ are known, you could find $theta$ such that $logtantheta=logfrac ba=log b-log a$, then $logsqrt{a^2+b^2}=log a+logsqrt{1+tan^2theta}$ and $logsqrt{1+tan^2theta}=logfrac{1}{cos theta}=-logcostheta$. There are many similar formulas.



            For geometric applications, sometimes versine and similar functions allow computing with greater precision while not adding too much computation. See for instance the haversine formula used to compute great circle distance (useful in navigation). The straightforward formula with arccosine has poor accuracy when the angle is small (the most common case), due to the fact that cosine is flat at $0$. However the formula with haversine is more accurate. To achieve the same, you would have to use $sin^2(theta/2)$ everywhere, which require more computations (but it's still reasonable with logarithms). Therefore, navigation table like Nories's nautical tables have an haversine table.



            Note that tables of logarithms are usually accurate up to 5 digits (some larger tables had 7 digits, some very special ones had better precision but were difficult to use : more digits = much more space on paper). Slide rules have roughly 3 digits of precision.



            All of this is rendered pretty useless with calculators, which have usually around 15 digits of precision and compute fast enough that we don't have to worry about speeding things up with extra functions.






            share|cite|improve this answer











            $endgroup$


















              6












              $begingroup$

              Those functions are much less used than before for one reason: the advent of electronic computers.



              Before that, one had to rely either on tables or on slide rules. Tables were usually table of logarithms, and they included the logarithms of trigonometric functions as well. The trigonometric functions were then useful not only for geometric applications, but also to simplify algebraic calculations with logarithms.



              For instance, to compute $logsqrt{a^2+b^2}$ when $log a$ and $log b$ are known, you could find $theta$ such that $logtantheta=logfrac ba=log b-log a$, then $logsqrt{a^2+b^2}=log a+logsqrt{1+tan^2theta}$ and $logsqrt{1+tan^2theta}=logfrac{1}{cos theta}=-logcostheta$. There are many similar formulas.



              For geometric applications, sometimes versine and similar functions allow computing with greater precision while not adding too much computation. See for instance the haversine formula used to compute great circle distance (useful in navigation). The straightforward formula with arccosine has poor accuracy when the angle is small (the most common case), due to the fact that cosine is flat at $0$. However the formula with haversine is more accurate. To achieve the same, you would have to use $sin^2(theta/2)$ everywhere, which require more computations (but it's still reasonable with logarithms). Therefore, navigation table like Nories's nautical tables have an haversine table.



              Note that tables of logarithms are usually accurate up to 5 digits (some larger tables had 7 digits, some very special ones had better precision but were difficult to use : more digits = much more space on paper). Slide rules have roughly 3 digits of precision.



              All of this is rendered pretty useless with calculators, which have usually around 15 digits of precision and compute fast enough that we don't have to worry about speeding things up with extra functions.






              share|cite|improve this answer











              $endgroup$
















                6












                6








                6





                $begingroup$

                Those functions are much less used than before for one reason: the advent of electronic computers.



                Before that, one had to rely either on tables or on slide rules. Tables were usually table of logarithms, and they included the logarithms of trigonometric functions as well. The trigonometric functions were then useful not only for geometric applications, but also to simplify algebraic calculations with logarithms.



                For instance, to compute $logsqrt{a^2+b^2}$ when $log a$ and $log b$ are known, you could find $theta$ such that $logtantheta=logfrac ba=log b-log a$, then $logsqrt{a^2+b^2}=log a+logsqrt{1+tan^2theta}$ and $logsqrt{1+tan^2theta}=logfrac{1}{cos theta}=-logcostheta$. There are many similar formulas.



                For geometric applications, sometimes versine and similar functions allow computing with greater precision while not adding too much computation. See for instance the haversine formula used to compute great circle distance (useful in navigation). The straightforward formula with arccosine has poor accuracy when the angle is small (the most common case), due to the fact that cosine is flat at $0$. However the formula with haversine is more accurate. To achieve the same, you would have to use $sin^2(theta/2)$ everywhere, which require more computations (but it's still reasonable with logarithms). Therefore, navigation table like Nories's nautical tables have an haversine table.



                Note that tables of logarithms are usually accurate up to 5 digits (some larger tables had 7 digits, some very special ones had better precision but were difficult to use : more digits = much more space on paper). Slide rules have roughly 3 digits of precision.



                All of this is rendered pretty useless with calculators, which have usually around 15 digits of precision and compute fast enough that we don't have to worry about speeding things up with extra functions.






                share|cite|improve this answer











                $endgroup$



                Those functions are much less used than before for one reason: the advent of electronic computers.



                Before that, one had to rely either on tables or on slide rules. Tables were usually table of logarithms, and they included the logarithms of trigonometric functions as well. The trigonometric functions were then useful not only for geometric applications, but also to simplify algebraic calculations with logarithms.



                For instance, to compute $logsqrt{a^2+b^2}$ when $log a$ and $log b$ are known, you could find $theta$ such that $logtantheta=logfrac ba=log b-log a$, then $logsqrt{a^2+b^2}=log a+logsqrt{1+tan^2theta}$ and $logsqrt{1+tan^2theta}=logfrac{1}{cos theta}=-logcostheta$. There are many similar formulas.



                For geometric applications, sometimes versine and similar functions allow computing with greater precision while not adding too much computation. See for instance the haversine formula used to compute great circle distance (useful in navigation). The straightforward formula with arccosine has poor accuracy when the angle is small (the most common case), due to the fact that cosine is flat at $0$. However the formula with haversine is more accurate. To achieve the same, you would have to use $sin^2(theta/2)$ everywhere, which require more computations (but it's still reasonable with logarithms). Therefore, navigation table like Nories's nautical tables have an haversine table.



                Note that tables of logarithms are usually accurate up to 5 digits (some larger tables had 7 digits, some very special ones had better precision but were difficult to use : more digits = much more space on paper). Slide rules have roughly 3 digits of precision.



                All of this is rendered pretty useless with calculators, which have usually around 15 digits of precision and compute fast enough that we don't have to worry about speeding things up with extra functions.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited 57 mins ago

























                answered 1 hour ago









                Jean-Claude ArbautJean-Claude Arbaut

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