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How were sine and cosine taught to you as anything other than ratios? I just assumed they were always taught as "opposite over hypotenuse" and "adjacent over hy

How were sine and cosine taught to you as anything other than ratios? I just assumed they were always taught as "opposite over hypotenuse" and "adjacent over hypotenuse" etc.

hungryforcodes 5y ago
They were clear to me intellectually, but the way he explained it made me SEE the ratios. Instead of just being abstract functions, there was a clear relationship between the angles in the triangle. After that I was able to reason about them, where as before they were just things I plugged into my calculator.
- mixmastamyk 5y ago
  Sometimes we are not ready for specific bits of information. i.e. brains have a "limited bandwidth." It's possible back in the old days the math teacher did mention it, but we couldn't understand it at the time because we had little sense of the subject. A common take on this phenomenon is discussed here occasionally: http://habitatchronicles.com/2004/04/you-cant-tell-people-anything/ http://habitatchronicles.com/2004/04/you-cant-tell-people-an... For a demonstration, go back and watch a good movie that requires focus, say three times in a row. Every viewing should uncover details you didn't notice the first time, because we can't take them all in at once. It's also why repetition is a good learning technique. Unfortunately few people have the patience to start over from the absolute beginning.
  - r00fus 5y ago
    This is very true. You are not the person you were in HS - everyone is a "Ship of Theseus" [1] therefore you can only guess at what HS version of you was like based on fuzzy remembrances. At your current age, you are more capable in some ways, less in others, and thus your learning capacity is different. [1] https://en.wikipedia.org/wiki/Ship_of_Theseus https://en.wikipedia.org/wiki/Ship_of_Theseus
mcbuilder 5y ago
I was originally taught the standard SOHCAHTOA in high school, and it didn't really click for me at all until I got to my undergrad introductory functions and graphs course. Near the end we got to Euler's forumla and our professor starting graphing e^{ix} in an extended complex plane. So apparently people learn things differently, as I prefer to work with identities and conceptual understanding rather than disconnected concepts.
yumraj 5y ago
That is how I learned too, however I believe the point of parent was that you can memorize it as "opposite over hypotenuse" and "base/adjacent over hypotenuse" OR you can have a realization that they are constant ratios in a right angled triangle. It's subtle, but I think it's a very valid point since I too never thought it in terms of ratio, merely in terms of here's the angle, this is what you need to find, you apply sin/cos/tan and get the answer.
tzs 5y ago
In high school any other way would probably be borderline insane. If you somehow managed to skip trigonometry in high school but still got into college calculus, then there are a few of other ways they might be defined. One way is to put off dealing with trig functions until you get to differential equations. Then sin is the unique solution of y'' + y = 0, y(0) = 0, y'(0) = 1, and cos is the unique solution of y'' + y = 0, y(0) = 1, y'(0) = 0. Another way is to define them axiomatically. These are the axioms for sin and cos: 1. They are both defined everywhere on the real line, 2. cos(0) = sin(pi/2) = 1, cos(pi) = -1, 3. cos(y-x) = cos(y) cos(x) + sin(y) sin(x) 4. for 0 < x < pi/2, 0 < cos(x) < sin(x)/x < 1/cos(x). From that you can deduce all the usual trig identities and the limits you need to do calculus with them. You still need to prove that functions satisfying those 4 axioms actually exist. One approach is to put that off until later when you've got more tools under you belt. For example once you do Taylor series, you can work out what they Taylor series would be for sin and cos if they exist. The functions defined by those Taylor series do exist, so then you just need to show that those functions satisfy the axioms.
- jacobolus 5y ago
  The reasonable ways to rigorously define sine and cosine functions (as functions) are: (1) (analytic continuations of) inverses of particular integrals (2) the sums of particular power series (3) solutions to a differential equation (4) as odd/even parts of the complex exponential, itself defined via (1), (2), or (3). Proving that these are all equivalent is a good exercise for a complex analysis student. But all of these require calculus and quite a lot of hand waving about real numbers and limits and infinite sums, especially in a high school context. And most of their applications (per se) are pretty advanced. At the high school level it’s probably best to explain that angle measure per se is a very tricky concept (an angle measure is a type of logarithm). We can get a very long way without ever having any explicit "angle measure" concept. E.g. there are no angle measures anywhere in Euclid’s Elements. (Angle measures come from astronomy and navigation.) We can look at the complex coordinates of 2D rotations (points on the complex unit circle) without ever trying to turn that into a single scalar-valued quantity. And should do that in a lot of circumstances where angle measures are currently used. In e.g. geometric simulations (computer graphics, robotics, computational geometry, ...) it’s best to avoid angle measures as much as possible, and stick to vector algebra.

How were sine and cosine taught to you as anything other than ratios? I just assumed they were always taught as "opposite over hypotenuse" and "adjacent over hypotenuse" etc.

hungryforcodes 5y ago

They were clear to me intellectually, but the way he explained it made me SEE the ratios. Instead of just being abstract functions, there was a clear relationship between the angles in the triangle. After that I was able to reason about them, where as before they were just things I plugged into my calculator.

mixmastamyk 5y ago
Sometimes we are not ready for specific bits of information. i.e. brains have a "limited bandwidth." It's possible back in the old days the math teacher did mention it, but we couldn't understand it at the time because we had little sense of the subject. A common take on this phenomenon is discussed here occasionally: http://habitatchronicles.com/2004/04/you-cant-tell-people-anything/ http://habitatchronicles.com/2004/04/you-cant-tell-people-an... For a demonstration, go back and watch a good movie that requires focus, say three times in a row. Every viewing should uncover details you didn't notice the first time, because we can't take them all in at once. It's also why repetition is a good learning technique. Unfortunately few people have the patience to start over from the absolute beginning.
- r00fus 5y ago
  This is very true. You are not the person you were in HS - everyone is a "Ship of Theseus" [1] therefore you can only guess at what HS version of you was like based on fuzzy remembrances. At your current age, you are more capable in some ways, less in others, and thus your learning capacity is different. [1] https://en.wikipedia.org/wiki/Ship_of_Theseus https://en.wikipedia.org/wiki/Ship_of_Theseus

mcbuilder 5y ago

I was originally taught the standard SOHCAHTOA in high school, and it didn't really click for me at all until I got to my undergrad introductory functions and graphs course. Near the end we got to Euler's forumla and our professor starting graphing e^{ix} in an extended complex plane. So apparently people learn things differently, as I prefer to work with identities and conceptual understanding rather than disconnected concepts.

yumraj 5y ago

That is how I learned too, however I believe the point of parent was that you can memorize it as "opposite over hypotenuse" and "base/adjacent over hypotenuse" OR you can have a realization that they are constant ratios in a right angled triangle. It's subtle, but I think it's a very valid point since I too never thought it in terms of ratio, merely in terms of here's the angle, this is what you need to find, you apply sin/cos/tan and get the answer.

tzs 5y ago

In high school any other way would probably be borderline insane. If you somehow managed to skip trigonometry in high school but still got into college calculus, then there are a few of other ways they might be defined. One way is to put off dealing with trig functions until you get to differential equations. Then sin is the unique solution of y'' + y = 0, y(0) = 0, y'(0) = 1, and cos is the unique solution of y'' + y = 0, y(0) = 1, y'(0) = 0. Another way is to define them axiomatically. These are the axioms for sin and cos: 1. They are both defined everywhere on the real line, 2. cos(0) = sin(pi/2) = 1, cos(pi) = -1, 3. cos(y-x) = cos(y) cos(x) + sin(y) sin(x) 4. for 0 < x < pi/2, 0 < cos(x) < sin(x)/x < 1/cos(x). From that you can deduce all the usual trig identities and the limits you need to do calculus with them. You still need to prove that functions satisfying those 4 axioms actually exist. One approach is to put that off until later when you've got more tools under you belt. For example once you do Taylor series, you can work out what they Taylor series would be for sin and cos if they exist. The functions defined by those Taylor series do exist, so then you just need to show that those functions satisfy the axioms.

jacobolus 5y ago
The reasonable ways to rigorously define sine and cosine functions (as functions) are: (1) (analytic continuations of) inverses of particular integrals (2) the sums of particular power series (3) solutions to a differential equation (4) as odd/even parts of the complex exponential, itself defined via (1), (2), or (3). Proving that these are all equivalent is a good exercise for a complex analysis student. But all of these require calculus and quite a lot of hand waving about real numbers and limits and infinite sums, especially in a high school context. And most of their applications (per se) are pretty advanced. At the high school level it’s probably best to explain that angle measure per se is a very tricky concept (an angle measure is a type of logarithm). We can get a very long way without ever having any explicit "angle measure" concept. E.g. there are no angle measures anywhere in Euclid’s Elements. (Angle measures come from astronomy and navigation.) We can look at the complex coordinates of 2D rotations (points on the complex unit circle) without ever trying to turn that into a single scalar-valued quantity. And should do that in a lot of circumstances where angle measures are currently used. In e.g. geometric simulations (computer graphics, robotics, computational geometry, ...) it’s best to avoid angle measures as much as possible, and stick to vector algebra.