How are Trignometric Ratios related to a circle?












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$begingroup$


How can an angle be compared to a circle?



tan means Tangent what does the ratio of Opposite side to Adjacent side have to do with the tangent of a circle?
Similarly, sec means Secant; How does ratio of hypotenuse to Adjacent side have to do with the secant of a circle?



Trigonometric ratios all apply only to a right angled triangle but still the names of the ratios suggest relations to entities related to circles.



I don't get the point of naming the ratios in such a way.






geometry trigonometry







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    I recommend watching this video <youtube.com/watch?v=6-tsSJx-P18>.
    $endgroup$
    – Gnumbertester
    Jan 25 at 15:05
















0












$begingroup$


How can an angle be compared to a circle?



tan means Tangent what does the ratio of Opposite side to Adjacent side have to do with the tangent of a circle?
Similarly, sec means Secant; How does ratio of hypotenuse to Adjacent side have to do with the secant of a circle?



Trigonometric ratios all apply only to a right angled triangle but still the names of the ratios suggest relations to entities related to circles.



I don't get the point of naming the ratios in such a way.






geometry trigonometry







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    I recommend watching this video <youtube.com/watch?v=6-tsSJx-P18>.
    $endgroup$
    – Gnumbertester
    Jan 25 at 15:05














0












0








0





$begingroup$


How can an angle be compared to a circle?



tan means Tangent what does the ratio of Opposite side to Adjacent side have to do with the tangent of a circle?
Similarly, sec means Secant; How does ratio of hypotenuse to Adjacent side have to do with the secant of a circle?



Trigonometric ratios all apply only to a right angled triangle but still the names of the ratios suggest relations to entities related to circles.



I don't get the point of naming the ratios in such a way.






geometry trigonometry







share|cite|improve this question









$endgroup$




How can an angle be compared to a circle?



tan means Tangent what does the ratio of Opposite side to Adjacent side have to do with the tangent of a circle?
Similarly, sec means Secant; How does ratio of hypotenuse to Adjacent side have to do with the secant of a circle?



Trigonometric ratios all apply only to a right angled triangle but still the names of the ratios suggest relations to entities related to circles.



I don't get the point of naming the ratios in such a way.






geometry trigonometry




geometry trigonometry






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share|cite|improve this question











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asked Jan 25 at 14:58









YourPalNuravYourPalNurav

1012




1012








  • 1




    $begingroup$
    I recommend watching this video <youtube.com/watch?v=6-tsSJx-P18>.
    $endgroup$
    – Gnumbertester
    Jan 25 at 15:05














  • 1




    $begingroup$
    I recommend watching this video <youtube.com/watch?v=6-tsSJx-P18>.
    $endgroup$
    – Gnumbertester
    Jan 25 at 15:05








1




1




$begingroup$
I recommend watching this video <youtube.com/watch?v=6-tsSJx-P18>.
$endgroup$
– Gnumbertester
Jan 25 at 15:05




$begingroup$
I recommend watching this video <youtube.com/watch?v=6-tsSJx-P18>.
$endgroup$
– Gnumbertester
Jan 25 at 15:05










3 Answers
3






active

oldest

votes


















1












$begingroup$


Trigonometric ratios all apply only to a right angled triangle




This statement is not historically accurate.
Historically, trigonometric ratios came from a circle.
The "triangle" part was a later idea.
See this answer for more details.



Somewhere during the history of teaching trigonometry, someone apparently got the idea that only certain triangles you can draw inside the circle are worth teaching to beginning students, and they decided to ignore the circle itself entirely.
I think this is a shame, because it later becomes a stumbling block if you get to the kind of mathematics where you need to take functions of any angle
(including negative angles and angles that "wrap around" multiple times),
not just angles between zero and a right angle.



A reasonably good definition of trigonometric functions can still be based on a circle,
for example, as in graphical representation of trig functions.
There are yet other definitions in higher mathematics that use neither circles nor triangles,
but the circle definition is the one that seems to be the source of the names of the functions.






share|cite|improve this answer











$endgroup$





















    0












    $begingroup$

    The length of a line segment tangential to a circle around a central angle is the tangent of that central angle. Look at this graph of the tangent function. enter image description here



    As you approach each new period, $frac{pi}{2}$, $frac{3pi}{2}$, etc, the value of tangent increases without bound, just as it would if you were to increase the angle $theta$ towards $frac{pi}{2}$ radians or $90$ degrees on this unit circle. enter image description here






    share|cite|improve this answer









    $endgroup$





















      0












      $begingroup$

      The origin of tangent is obvious: the function denotes the length intercepted by the (extended) radius along the tangent at point $(1,0)$.






      share|cite|improve this answer









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        3 Answers
        3






        active

        oldest

        votes








        3 Answers
        3






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes









        1












        $begingroup$


        Trigonometric ratios all apply only to a right angled triangle




        This statement is not historically accurate.
        Historically, trigonometric ratios came from a circle.
        The "triangle" part was a later idea.
        See this answer for more details.



        Somewhere during the history of teaching trigonometry, someone apparently got the idea that only certain triangles you can draw inside the circle are worth teaching to beginning students, and they decided to ignore the circle itself entirely.
        I think this is a shame, because it later becomes a stumbling block if you get to the kind of mathematics where you need to take functions of any angle
        (including negative angles and angles that "wrap around" multiple times),
        not just angles between zero and a right angle.



        A reasonably good definition of trigonometric functions can still be based on a circle,
        for example, as in graphical representation of trig functions.
        There are yet other definitions in higher mathematics that use neither circles nor triangles,
        but the circle definition is the one that seems to be the source of the names of the functions.






        share|cite|improve this answer











        $endgroup$


















          1












          $begingroup$


          Trigonometric ratios all apply only to a right angled triangle




          This statement is not historically accurate.
          Historically, trigonometric ratios came from a circle.
          The "triangle" part was a later idea.
          See this answer for more details.



          Somewhere during the history of teaching trigonometry, someone apparently got the idea that only certain triangles you can draw inside the circle are worth teaching to beginning students, and they decided to ignore the circle itself entirely.
          I think this is a shame, because it later becomes a stumbling block if you get to the kind of mathematics where you need to take functions of any angle
          (including negative angles and angles that "wrap around" multiple times),
          not just angles between zero and a right angle.



          A reasonably good definition of trigonometric functions can still be based on a circle,
          for example, as in graphical representation of trig functions.
          There are yet other definitions in higher mathematics that use neither circles nor triangles,
          but the circle definition is the one that seems to be the source of the names of the functions.






          share|cite|improve this answer











          $endgroup$
















            1












            1








            1





            $begingroup$


            Trigonometric ratios all apply only to a right angled triangle




            This statement is not historically accurate.
            Historically, trigonometric ratios came from a circle.
            The "triangle" part was a later idea.
            See this answer for more details.



            Somewhere during the history of teaching trigonometry, someone apparently got the idea that only certain triangles you can draw inside the circle are worth teaching to beginning students, and they decided to ignore the circle itself entirely.
            I think this is a shame, because it later becomes a stumbling block if you get to the kind of mathematics where you need to take functions of any angle
            (including negative angles and angles that "wrap around" multiple times),
            not just angles between zero and a right angle.



            A reasonably good definition of trigonometric functions can still be based on a circle,
            for example, as in graphical representation of trig functions.
            There are yet other definitions in higher mathematics that use neither circles nor triangles,
            but the circle definition is the one that seems to be the source of the names of the functions.






            share|cite|improve this answer











            $endgroup$




            Trigonometric ratios all apply only to a right angled triangle




            This statement is not historically accurate.
            Historically, trigonometric ratios came from a circle.
            The "triangle" part was a later idea.
            See this answer for more details.



            Somewhere during the history of teaching trigonometry, someone apparently got the idea that only certain triangles you can draw inside the circle are worth teaching to beginning students, and they decided to ignore the circle itself entirely.
            I think this is a shame, because it later becomes a stumbling block if you get to the kind of mathematics where you need to take functions of any angle
            (including negative angles and angles that "wrap around" multiple times),
            not just angles between zero and a right angle.



            A reasonably good definition of trigonometric functions can still be based on a circle,
            for example, as in graphical representation of trig functions.
            There are yet other definitions in higher mathematics that use neither circles nor triangles,
            but the circle definition is the one that seems to be the source of the names of the functions.







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Jan 25 at 16:24

























            answered Jan 25 at 16:16









            David KDavid K

            55.2k344120




            55.2k344120























                0












                $begingroup$

                The length of a line segment tangential to a circle around a central angle is the tangent of that central angle. Look at this graph of the tangent function. enter image description here



                As you approach each new period, $frac{pi}{2}$, $frac{3pi}{2}$, etc, the value of tangent increases without bound, just as it would if you were to increase the angle $theta$ towards $frac{pi}{2}$ radians or $90$ degrees on this unit circle. enter image description here






                share|cite|improve this answer









                $endgroup$


















                  0












                  $begingroup$

                  The length of a line segment tangential to a circle around a central angle is the tangent of that central angle. Look at this graph of the tangent function. enter image description here



                  As you approach each new period, $frac{pi}{2}$, $frac{3pi}{2}$, etc, the value of tangent increases without bound, just as it would if you were to increase the angle $theta$ towards $frac{pi}{2}$ radians or $90$ degrees on this unit circle. enter image description here






                  share|cite|improve this answer









                  $endgroup$
















                    0












                    0








                    0





                    $begingroup$

                    The length of a line segment tangential to a circle around a central angle is the tangent of that central angle. Look at this graph of the tangent function. enter image description here



                    As you approach each new period, $frac{pi}{2}$, $frac{3pi}{2}$, etc, the value of tangent increases without bound, just as it would if you were to increase the angle $theta$ towards $frac{pi}{2}$ radians or $90$ degrees on this unit circle. enter image description here






                    share|cite|improve this answer









                    $endgroup$



                    The length of a line segment tangential to a circle around a central angle is the tangent of that central angle. Look at this graph of the tangent function. enter image description here



                    As you approach each new period, $frac{pi}{2}$, $frac{3pi}{2}$, etc, the value of tangent increases without bound, just as it would if you were to increase the angle $theta$ towards $frac{pi}{2}$ radians or $90$ degrees on this unit circle. enter image description here







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered Jan 25 at 15:19









                    GnumbertesterGnumbertester

                    670114




                    670114























                        0












                        $begingroup$

                        The origin of tangent is obvious: the function denotes the length intercepted by the (extended) radius along the tangent at point $(1,0)$.






                        share|cite|improve this answer









                        $endgroup$


















                          0












                          $begingroup$

                          The origin of tangent is obvious: the function denotes the length intercepted by the (extended) radius along the tangent at point $(1,0)$.






                          share|cite|improve this answer









                          $endgroup$
















                            0












                            0








                            0





                            $begingroup$

                            The origin of tangent is obvious: the function denotes the length intercepted by the (extended) radius along the tangent at point $(1,0)$.






                            share|cite|improve this answer









                            $endgroup$



                            The origin of tangent is obvious: the function denotes the length intercepted by the (extended) radius along the tangent at point $(1,0)$.







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered Jan 25 at 16:32









                            Yves DaoustYves Daoust

                            131k676229




                            131k676229






























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