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> Trig is a bunch of arbitrary formulas if you don't have the calculus behind them. Why do you claim this? You don't have to have seen calculus to appreciate h

> Trig is a bunch of arbitrary formulas if you don't have the calculus behind them. Why do you claim this? You don't have to have seen calculus to appreciate how trig functions are defined, how to manipulate them, or how to use them in applications. As a math professor, I personally like the fact that we teach trig and exponential/logarithmic functions before calculus. They are (as you well know) exceedingly rich examples which illustrate why calculus is interesting and useful, and knowing them already enables the student to study calculus without excessive digressions.

stephencanon 11y ago
sin and cos are the natural basis of solutions of y'' + y = 0. How do you define them?
- tarentel 11y ago
  Ratios in a right triangle perhaps?
  - threatofrain 11y ago
    How did you get that information? How did you, in a non-magical way, go from information about an angle to information about a ratio? If students don't know this information, then perhaps they are studying applications. So, what applications are students taught in typical trigonometric texts? Periodic behavior perhaps? Like sound? Only perhaps a brief blurb in the text that application is even possible. Perhaps they look at something about an incline plane. It is unlikely that they will touch projectiles. It appears that trigonometry is there to give students some sense of mild comfort for future work in physics or engineering. This makes me think, "Why not statistics instead?"
    - AnimalMuppet 11y ago
      > How did you, in a non-magical way, go from information about an angle to information about a ratio? By having a right triangle? The rest of your post seems to show that you want trig to be about periodic behavior, when it really is about triangles. That's what trigonometry means - measuring triangles. Yes, trig has applications to periodic behavior, projectiles, differential equations, inclined planes, and all kinds of other stuff. But the point of a trig class is not to teach the applications. The point is to teach the tools, and maybe touch on the applications.
      - btilly 11y ago
        The problem is this. Using compass and ruler constructions there is a set of angles you can construct, and you can calculate sin and cos for those angles. You can even write the values for those out explicitly. However no part of this construction sheds light on how to find sin and cos for angles that you don't know how to construct. Or even gives good intuition that no matter how you do it, you can define it in a way that makes sense for all angles. In fact we draw a picture, people look at it, and their intuition tells them that things will work out. Very few students will notice the logical gaps. But to close the logical gaps, you need to start with Calculus first, and then derive trig formulas from that. (Yes, I'm aware of the history here. Euclid presented trig reasonably rigorously a very long time before Calculus. Newton invented Calculus in the 1600s, and then used it as a heuristic to figure out answers that he then rederived using trig in The Principia. Leibniz reinvented Calculus in part based on inspiration from Newton's work. None of this was made formally correct until the late 1800s.) (I have no opinion on pedagogical arguments about which is best to present first. I believe that we present trig first as a holdover from a curriculum where The Elements was the standard textbook until very recently.)
        AnimalMuppet 11y ago
        Well... you can use the half-angle formulas and the angle addition formulas to calculate sin and cos for angles that are arbitrarily close to the ones that you want. Add to that the idea that sin and cos must be continuous (I consider that intuitively obvious from a unit circle, but I don't know how to make that argument rigorous), and you can start to interpolate. You can in fact use these methods to calculate sin and cos for an arbitrary angle to any desired degree of precision... if you have the patience. It will be shorter to use the series derived from calculus, I'll admit.
        btilly 11y ago
        I do not believe that there is an argument for continuity without starting with Calculus. Certainly starting from ruler and compass constructions it is not obvious. That said, if you have enough Calculus to define how to measure the arclength of a segment of the circle, you can quickly prove that sin and cos in radians exist, have a nice power series, and so on. It is like x^y with x positive. We can manually define it every rational y. But the easiest way to get a rigorous and straightforward definition is to prove the algebraic properties of the integral of 1/x, use that to define the logarithm, define its inverse function to be the exponential, prove its algebraic properties, then define x^y as e^(y*log(x)). And it all just works.
    - replicant 11y ago
      I think I am missing something because, I am unable to see why it is huge burden to introduce sine and cosine without their rigorous definition. At which age, are students taught trigonometry? And what does a course on trigonometry covers? What would you think they would be able to do without it? When we were introduced the sine and the cosine function, we were already familiar with Thales theorem, so therefore we could show that this ratio was a constant. I am quite sure historically as well sine and cosine predate the more formal construction of those functions, be it as a series, solution of an ODE or inverse of arc sin (and this defined as an integral)...
      - threatofrain 11y ago
        I think I wasn't clear in saying that I believe the current pedagogical value of trigonometry is in giving students a brief familiarity with the trig functions when they see it again in the context of physics or engineering. Or standardized testing. I think those are the likely scenarios where students are going to be seeing relevance in trigonometry. What other foundation or learning pathway do you see trigonometry serving as? Somebody else mentioned that it gives students a sense of applications, so they know that calculus is not for nothing. So then I question: what applications? And I pose, how about statistics?
        east2west 11y ago
        It may be because of your math education but I was introduced to trigonometry in junior high in China. I had, and my classmates had, no trouble understanding them as functions of angles coming from ratios. It is the glue that binds circles and triangles and squares and ... . By high school analytic geometry greatly expanded their scope and use. This is the problem I saw in American high school when I moved to US: shallow introduction to mathematical topics made them vapid and jejune. Ancient Greeks were enthralled by trigonometry, ancient Egyptians built Pyramids and ancient Chinese built great dams with trigonometry. Calling it practically useless and pedagogically only useful as a prep for calculus is going too far.
    - Someone 11y ago
      "So, what applications are students taught in typical trigonometric texts?" "If a pyramid is 250 cubits high and the side of its base 360 cubits long, what is its seked?" (http://en.m.wikipedia.org/wiki/Rhind_Mathematical_Papyrus#Pyramids http://en.m.wikipedia.org/wiki/Rhind_Mathematical_Papyrus#Py...) Well, maybe not that typical, but it is an example without any periodicity in sight.
    - stan_rogers 11y ago
      My introduction was through model rocketry. Somewhere along the line, Al-Biruni's method for finding the radius of the earth was brought up. That was in the early '70s, in what would be "junior high" in the US (elementary school in my part of Canada). It was always about right triangles, not periodic functions, at the beginning, which makes a whole lot of sense - trigonometry was both useful and used for a whole lot of years before calculus was invented. And like logarithms in the pre-scientific-calculator days, there was a point where one turned to tables for practical reasons without thinking of the table values as "magicical" - we were taught how to calculate intermediate values to the limits of practicality. Is there any practical sense (a sense that would be useful for people who would be entering the trades track) in being able to calculate much more accurately than you can measure angles?
- johnaspden 11y ago
  coordinates of rotaty-thing
- monochromatic 11y ago
  Infinite sums are as clean as anything.
tokenrove 11y ago
I don't know, I agree with the parent. I hated trig in my first encounters with it; the thing that made it practical was writing videogames and graphics demos, and the thing that made it interesting was calculus. For many people, it's the last math they're taught, they don't get the applications, and it's no wonder they don't hunger for more.

stephencanon 11y ago

sin and cos are the natural basis of solutions of y'' + y = 0. How do you define them?

tarentel 11y ago
Ratios in a right triangle perhaps?
- threatofrain 11y ago
  How did you get that information? How did you, in a non-magical way, go from information about an angle to information about a ratio? If students don't know this information, then perhaps they are studying applications. So, what applications are students taught in typical trigonometric texts? Periodic behavior perhaps? Like sound? Only perhaps a brief blurb in the text that application is even possible. Perhaps they look at something about an incline plane. It is unlikely that they will touch projectiles. It appears that trigonometry is there to give students some sense of mild comfort for future work in physics or engineering. This makes me think, "Why not statistics instead?"
  - AnimalMuppet 11y ago
    > How did you, in a non-magical way, go from information about an angle to information about a ratio? By having a right triangle? The rest of your post seems to show that you want trig to be about periodic behavior, when it really is about triangles. That's what trigonometry means - measuring triangles. Yes, trig has applications to periodic behavior, projectiles, differential equations, inclined planes, and all kinds of other stuff. But the point of a trig class is not to teach the applications. The point is to teach the tools, and maybe touch on the applications.
    - btilly 11y ago
      The problem is this. Using compass and ruler constructions there is a set of angles you can construct, and you can calculate sin and cos for those angles. You can even write the values for those out explicitly. However no part of this construction sheds light on how to find sin and cos for angles that you don't know how to construct. Or even gives good intuition that no matter how you do it, you can define it in a way that makes sense for all angles. In fact we draw a picture, people look at it, and their intuition tells them that things will work out. Very few students will notice the logical gaps. But to close the logical gaps, you need to start with Calculus first, and then derive trig formulas from that. (Yes, I'm aware of the history here. Euclid presented trig reasonably rigorously a very long time before Calculus. Newton invented Calculus in the 1600s, and then used it as a heuristic to figure out answers that he then rederived using trig in The Principia. Leibniz reinvented Calculus in part based on inspiration from Newton's work. None of this was made formally correct until the late 1800s.) (I have no opinion on pedagogical arguments about which is best to present first. I believe that we present trig first as a holdover from a curriculum where The Elements was the standard textbook until very recently.)
      - AnimalMuppet 11y ago
        Well... you can use the half-angle formulas and the angle addition formulas to calculate sin and cos for angles that are arbitrarily close to the ones that you want. Add to that the idea that sin and cos must be continuous (I consider that intuitively obvious from a unit circle, but I don't know how to make that argument rigorous), and you can start to interpolate. You can in fact use these methods to calculate sin and cos for an arbitrary angle to any desired degree of precision... if you have the patience. It will be shorter to use the series derived from calculus, I'll admit.
        btilly 11y ago
        I do not believe that there is an argument for continuity without starting with Calculus. Certainly starting from ruler and compass constructions it is not obvious. That said, if you have enough Calculus to define how to measure the arclength of a segment of the circle, you can quickly prove that sin and cos in radians exist, have a nice power series, and so on. It is like x^y with x positive. We can manually define it every rational y. But the easiest way to get a rigorous and straightforward definition is to prove the algebraic properties of the integral of 1/x, use that to define the logarithm, define its inverse function to be the exponential, prove its algebraic properties, then define x^y as e^(y*log(x)). And it all just works.
  - replicant 11y ago
    I think I am missing something because, I am unable to see why it is huge burden to introduce sine and cosine without their rigorous definition. At which age, are students taught trigonometry? And what does a course on trigonometry covers? What would you think they would be able to do without it? When we were introduced the sine and the cosine function, we were already familiar with Thales theorem, so therefore we could show that this ratio was a constant. I am quite sure historically as well sine and cosine predate the more formal construction of those functions, be it as a series, solution of an ODE or inverse of arc sin (and this defined as an integral)...
    - threatofrain 11y ago
      I think I wasn't clear in saying that I believe the current pedagogical value of trigonometry is in giving students a brief familiarity with the trig functions when they see it again in the context of physics or engineering. Or standardized testing. I think those are the likely scenarios where students are going to be seeing relevance in trigonometry. What other foundation or learning pathway do you see trigonometry serving as? Somebody else mentioned that it gives students a sense of applications, so they know that calculus is not for nothing. So then I question: what applications? And I pose, how about statistics?
      - east2west 11y ago
        It may be because of your math education but I was introduced to trigonometry in junior high in China. I had, and my classmates had, no trouble understanding them as functions of angles coming from ratios. It is the glue that binds circles and triangles and squares and ... . By high school analytic geometry greatly expanded their scope and use. This is the problem I saw in American high school when I moved to US: shallow introduction to mathematical topics made them vapid and jejune. Ancient Greeks were enthralled by trigonometry, ancient Egyptians built Pyramids and ancient Chinese built great dams with trigonometry. Calling it practically useless and pedagogically only useful as a prep for calculus is going too far.
  - Someone 11y ago
    "So, what applications are students taught in typical trigonometric texts?" "If a pyramid is 250 cubits high and the side of its base 360 cubits long, what is its seked?" (http://en.m.wikipedia.org/wiki/Rhind_Mathematical_Papyrus#Pyramids http://en.m.wikipedia.org/wiki/Rhind_Mathematical_Papyrus#Py...) Well, maybe not that typical, but it is an example without any periodicity in sight.
  - stan_rogers 11y ago
    My introduction was through model rocketry. Somewhere along the line, Al-Biruni's method for finding the radius of the earth was brought up. That was in the early '70s, in what would be "junior high" in the US (elementary school in my part of Canada). It was always about right triangles, not periodic functions, at the beginning, which makes a whole lot of sense - trigonometry was both useful and used for a whole lot of years before calculus was invented. And like logarithms in the pre-scientific-calculator days, there was a point where one turned to tables for practical reasons without thinking of the table values as "magicical" - we were taught how to calculate intermediate values to the limits of practicality. Is there any practical sense (a sense that would be useful for people who would be entering the trades track) in being able to calculate much more accurately than you can measure angles?
johnaspden 11y ago
coordinates of rotaty-thing
monochromatic 11y ago
Infinite sums are as clean as anything.

tokenrove 11y ago

I don't know, I agree with the parent. I hated trig in my first encounters with it; the thing that made it practical was writing videogames and graphics demos, and the thing that made it interesting was calculus. For many people, it's the last math they're taught, they don't get the applications, and it's no wonder they don't hunger for more.