This is a bit of simple knowledge which is sadly unbeknownst to even most math majors and educators. Often, students are taught in calculus that cosines (and hence other trig functions) are computed with Taylor series, which is not really correct. In fact they use CORDIC, a highly optimized algorithm.
But CORDIC is based on repeated use of the sum formula, and a simple version of this can be taught without calculus. Just notice that:
cos(2x) = 2 cos(x)^2 - 1
Then for small enough x, you have (can be shown by drawing, but kind of annoying):
cos(x) ≈ 1 - x^2 / 2
Or rigorously: 1 - x^2 < cos(x)^2 < 1 / (x^2 + 1)
So divide x by 2 until you get a small number (x < 0.1 is usually good enough), use the quadratic approximation, and repeatedly apply the doubling formula. CORDIC uses a similar iteration based on the sum formula. There is no need to wait for Taylor series, limits, convergence tests, etc. You can even bound the error term!
I'm not entirely sure this would work (i.e. improve understanding), though — I have not taught high schoolers before. And there is a decent bit of work required.
I never tried a random angle like 13° in my head, but I figured it was kind of on a scale between the “round” numbers which I knew, which would give a good sense of the actual number, even if not accurate enough.
The point is, the unit circle explained the figures to me instead of them being just some magic numbers.
Likewise, there was an actual explanation of concepts like cos and sin in that they are the names for relationships between angles/sides (again, I can’t remember now, but it made sense), rather than them just being tools you’re told to use when.
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1. Although when I think about educational malpractice, I remember a program I was teaching in where the director asked me about why the numerator and denominator were called what they were and I explained the denominator identified what kind of fraction (and made the analogy to denominations of currency) and the numerator was how many (the “number”) of that kind of fraction. She then, that same day, gave a talk to the assembled teachers saying that she’d asked me and I’d said that’s just what they were called. I nearly stood up and screamed in anger. If I hadn’t desperately needed the money, I would have quit on the spot.
We were simply taught which equation to use for particular scenarios and how to use calculators with particular equations.
When you bought it up, I do recall the connection being used by Open University to explain things, which I also found a bit thrilling to learn.
Although only anecdotal evidence, I have asked a few South African colleagues over the years about the unit circle and it seems like none of them were taught it during school, so it seems like it's just not in the curriculums over here, which I feel is a tragedy.
Here’s how to create this. Draw a unit circle centered on the origin. Draw a ray from the origin where θ will be the angle measured from the positive x axis. You know sin and cos already: you draw a perpendicular from the x axis to where the ray intersects the circle. sinθ is the distance from the origin to where your perpendicular hits the x axis and cosθ is the distance from the x axis to where the perpendicular hits the circle.
Now, draw another line perpendicular from the x axis at (1,0) to your ray (tangent to the circle). Let’s call the point where this line hits your ray T. The distance from T to (1,0) is tanθ and the distance from the origin (0,0) to T is secθ. (As an added bonus, you can easily see that 1+tan²θ = sec²θ). We can do the same process, drawing a tangent line from (0,1) instead of (1,0) to get cotθ and cscθ. You can do some simple math with similar triangles to get familiar formulae like tanθ=sinθ/cosθ etc.
Explaining signs of the functions outside the first quadrant is left as an exercise to the reader.
I'm not a mathematician, but that sounds like poor teaching to me. Even if it's not part of the programme, learning the unit circle takes literally a few minutes and is an invaluable tool afterward. I don't know if it's possible to develop an intuition for trigonometry without learning the unit circle.
It also has a downside... The British Empire depended on robust trigonometrical education. In those days, you couldn't efficiently navigate the world's oceans without it and they needed a steady supply of ship's masters for the thousands of ships that made the empire work. Earth is the unit circle.
What exactly is the downside and for whom?
> The British Empire depended on robust trigonometrical education [...] and they needed a steady supply of ship's masters
The British Empire was already collapsing rapidly when the Open University was founded in 1969. I doubt that ensuring a supply of ship masters was a motivation in choosing to include trigonometry in the syllabus.
I asked the teacher where the numbers which were spat out by the calculator came from and was only told “in my day we didn’t have calculators and had to lookup the answers from a table on a book!”.
Which was such a thoroughly disappointing response and I hated trigonometry since it was mostly about memorisation rather than applying logic from understanding.
About 10 years later I did a math course through the UK’s Open University and the text book taught trigonometry through explaining the unit circle.
It made me so happy to finally understand and see the actual logic as opposed to punching numbers into a calculator, but I was also sad and a bit upset the unit circle was not taught in high school.