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?