4 ms·

>> What struck me as odd when I was an undergraduate, and still strikes me to this day, is that none of these are the obvious trigonometric definitions about th

>> What struck me as odd when I was an undergraduate, and still strikes me to this day, is that none of these are the obvious trigonometric definitions about the opposite and adjacent sides of a right triangle. I had the less common experience of learning some graphics programming prior to taking a trig class in school. "How do I draw a circle?" You sweep an angle from 0 to 360 (or 2*pi) and use sin() and cos() to get points on the unit circle, then multiply by your radius and plot them. For me the notion of sin and cos being coordinates of points on a unit circle was natural. Later when I had trig class it seemed really weird to define these functions as ratios of sides of a triangle. In particular you never talk about a side length as being negative. So no, it's not universal that the triangle definitions seem more obvious or make more intuitive sense. I think this is important for people to understand. What seems easy or natural can depend on how your particular tree of knowledge was constructed.

vlowrian 3y ago
In fact, sine and cosine never really made sense to me, until one day, I saw an animation similar to this one: https://youtu.be/Q55T6LeTvsA?si=7fiBxFVMu67tb3NZ https://youtu.be/Q55T6LeTvsA?si=7fiBxFVMu67tb3NZ (without the overly dramatic soundtrack). Before that it was just some weird function I needed to calculate angles in triangles…
- jimmySixDOF 3y ago
  Yes this article really should have animations to get its point across. Seems like such a natural fit while going through the effort of writing it and much less engaging as a result.
philomath_mn 3y ago
That's too bad -- I am pretty sure my trig class introduced unit circles and the triangle relationships at around the same time. The unit circle with key points labeled is still my go-to doodle; it is a beautiful set of concepts.
Sharlin 3y ago
I remember wondering what these strange "sin" and "cos" functions were in Delphi 2 or something – it was ~1997 and I couldn’t just google it – and tried to find a pattern in the seemingly arbitrary output values. I don’t think trying to plot them as y=f(x) crossed my mind, not at first anyway, but instead I somehow ended up trying to plot x=k sin(t), y=k cos(t) and was amazed when I realized that they traced a circle on the screen. Only later I learned in school that they had anything to do with triangles. To me they were functions for emitting circle coordinates first and foremost.
kens 3y ago
My experience was similar. I never really understood the point of sin and cos until I used them to draw circles on a TRS-80. And then it's like "Oh, they are very useful and straightforward functions." And then it made sense why you'd divide a triangle's side length by the hypotenuse, since that's just scaling by the radius.

vlowrian 3y ago

In fact, sine and cosine never really made sense to me, until one day, I saw an animation similar to this one: https://youtu.be/Q55T6LeTvsA?si=7fiBxFVMu67tb3NZ https://youtu.be/Q55T6LeTvsA?si=7fiBxFVMu67tb3NZ (without the overly dramatic soundtrack). Before that it was just some weird function I needed to calculate angles in triangles…

jimmySixDOF 3y ago
Yes this article really should have animations to get its point across. Seems like such a natural fit while going through the effort of writing it and much less engaging as a result.

philomath_mn 3y ago

That's too bad -- I am pretty sure my trig class introduced unit circles and the triangle relationships at around the same time. The unit circle with key points labeled is still my go-to doodle; it is a beautiful set of concepts.

Sharlin 3y ago

I remember wondering what these strange "sin" and "cos" functions were in Delphi 2 or something – it was ~1997 and I couldn’t just google it – and tried to find a pattern in the seemingly arbitrary output values. I don’t think trying to plot them as y=f(x) crossed my mind, not at first anyway, but instead I somehow ended up trying to plot x=k sin(t), y=k cos(t) and was amazed when I realized that they traced a circle on the screen. Only later I learned in school that they had anything to do with triangles. To me they were functions for emitting circle coordinates first and foremost.

kens 3y ago

My experience was similar. I never really understood the point of sin and cos until I used them to draw circles on a TRS-80. And then it's like "Oh, they are very useful and straightforward functions." And then it made sense why you'd divide a triangle's side length by the hypotenuse, since that's just scaling by the radius.