SOH CAH TOA vs. The Winding Function



I’m currently in my student-teacher placement at a local high-school and I find myself teaching trigonometry.  This is fun, but I’m expected to teach “the SOH CAH TOA” method, which is something that my teachers when I was in high-school (at Massey) were dead set against. Their reason was that using SOH CAH TOA does not create actual mathematical understanding of what sine and cosine are.

They are functions. They take numbers from the reals and map them to numbers in-between -1 and 1.

Using “SOH CAH TOA,” so say the teachers at Massey, creates the mistaken idea that the trigonometric functions are simply ratios of side lengths of right triangles. They use the (possibly) more complex concept of “the winding function” to explain how to solve right triangles. Imagine a unit circle centred at the origin. Extend a ray from the origin at an angle θ from the x-axis. The point where that ray intersects the unit circle will be (cosθ, sinθ). Now we know that we can use what we know about similar triangles to scale up or down as we wish.

This method does not hide the reality that sin and cos are functions and not ratios, and it uses previously learned knowledge from similar triangles to explain how to solve right angled triangles. However, most teachers would say that this definition requires some understanding of what a function is, but that concept is not taught until Grade 11. Also going any further in depth requires Grade 10s to know the equation for a circle, which has it’s own difficulties, and again is a Grade 11 topic.

How do you understand sin and cos? Does it really matter that Grade 10s are not getting a full understanding of cos and sin, considering that for most problems they will be required to solve the SOH CAH TOA method is perfectly adequate? What do you think dear readers?

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16 thoughts on “SOH CAH TOA vs. The Winding Function

  1. The idea is to teach abstraction, and not memorization. The point isn’t to simply explain what the functions mean, but to present a more flexible, and thus powerful, way of thinking.

  2. Dear Reg a few ideas

    1) Equations of circles are on the grade 10 curriculum.

    2) The most compelling argument for the use of the winding function is the understanding of trigonometry is the connection between a line and a curve.

    3) the definition of an angle on the winding function which you did not bring up gives a reality to Real number ( radian) measure of an angle.
    The amount of rotation about the origin from W(0) to W(theta) is theta. This makes the domain,the arc length and the angle all equal thus connecting the real numbers and angles.

    I hope that this helps but it is certainly too late for your lesson in May

    JBW

  3. Dear Reg a few of ideas

    1) Equations of circles are on the grade 10 curriculum.

    2) The most compelling argument for the use of the winding function is the understanding of trigonometry is the connection between a line and a curve.

    3) the definition of an angle on the winding function which you did not bring up gives a reality to Real number ( radian) measure of an angle.
    The amount of rotation about the origin from W(0) to W(theta) is theta. This makes the domain,the arc length and the angle all equal thus connecting the real numbers and angles.

    I hope that this helps but it is certainly too late for your lesson in May

    You do not show the domain in your diagram.

    JBW

    • Thanks a lot sir. It was strange having to teach it in a way that seemed backward and foreign to me. There was an idea at Riverside that the kids could not handle it the “smart” way. When I mentioned Massey, they would quickly point out that “these are not Massey students.” They seemed to feel that the students that they were teaching were somehow inherently inferior to ones at Massey. There was a bias against Massey. An assumption that teachers had it easy there and all the students were good. When I told them that I volunteered there they dismissed my experience, saying that “Massey isn’t real” or “It’s all different here.” I should have mentioned that I was working in Mr. Braithwaite’s Math for Everyday Life class but I was busy being taken aback.
      But, this made me think about Massey and how it could not have miraculously become a great school overnight. That it takes teachers devoted to fostering real understanding and a pro-learning environment to cultivate students that care. Thank you sir, for being one of those teachers, for pushing students to struggle through difficult concepts and problems, and for insisting that this is how teaching should be done. I never had the honour of being in your class, but I’m proud of the education I had at Massey. I learned, not just curriculum, but also some of the best lessons in how to be a teacher.

      Keep your epsilons positive,

      Reg

  4. Reg: I teach the winding function or unit circle to my 11’s and 12’s. So I don’t know where you got the idea that they can’t handle the “smart” way. The real reason it’s not taught in gr. 10 is it is not part of the curriculum and SOH CAH TOA is. You only deal with real-life examples that have angles less than 180 degrees so you don’t need to get into 3rd and 4th quadrant answers.
    You only dealt with one teacher at Riverside and as a math teacher you should know that is not a representative sample.

    Rachelle Leonard (Riverside Secondary math teacher)

    • Mrs. Leonard!! is that really you? it’s mina (your math student from Lowe). one xmas of about 10 years ago i visited you at Riverside school. do you still remember?

      if that’s really you, could you please reply?

      (I apologize to the Blog owner for ‘hijacking’ this thread, for I am very excited if indeed I have found my teacher again!)

  5. This is fascinating to me because I teach Trig at a high school where our AP Calculus teacher believes strongly that memorizing points on a unit circle is a terrible way to learn the trigonometric functions of common angles. Although we use the circle a great deal for many things (visualizing radians compared to degrees, finding sin and cos of quadrantal angles, knowing the signs of the quadrants, seeing how the graphs relate to the functions), when it comes to finding the sine of something like 210 degrees, the students are absolutely not allowed to do that on a unit circle. They need to know that it has the same sine as a 30 degree angle, other than the positive or negative, and then they need to know the sine of 30 degrees by use of a 30-60-90 triangle. This is considered the “smart” way, rather than just looking at a unit circle and finding 210 degrees. The students are actually forbidden to sketch the unit circle for a case like this as she does not believe it leads to understanding, just memorization.

    • At Massey you’re never taught to memorize all the points you only need to memorize 3 to know the whole circle W(0), W(pi/6), W(pi/4) that’s how we’re taught to know all the points the use in the function is like you said, in tackling contest, abstract problems, and proofs drawing a winding function helps draw connections between angular and spatial quantities as Dr. White pointed out which is the most important conceptual reason to teach the Winding Function in depth rather than SOH CAH TOA which is merely an acronym.

  6. yeahh man teachers at massey hate soh cah toa -_- !!
    i had to learn the winding function..which i didnt really learn :P

  7. I want to learn the winding function rather than soh cah toa but by the cirriculum of my school, it’s an absolute necessity. Does anyone have any links or websites where I can learn it?

  8. I’m not a Massey student, but I am a member of their math team for ARML.

    The Winding Function makes the concepts of trigonometry much more real and powerful. Before, we have different, arbitrarily set numbers: now we really get the depth of thinking of those who first investigated trigonometry.

    That is much more important than memorizing a stupid little mnemonic called SOH CAH TOA.

  9. Wow, I go to massey too! We are currently learning this and I believe SOHCAHTOA was very useful. This is just a simpler and faster way of figuring out the math. Time is precious, so the more you save the better.

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