I often tutor high school and undergraduate students, and I’ve noticed that those with limited exposure to trigonometry initially struggle to recall the standard sine and cosine values. They usually remember the key angles in the first quadrant (0°, 30°, 45°, 60°, 90°) and can identify corresponding angles in the other quadrants, but they often complain about the difficulty of memorizing the whole table.
A mnemonic I suggest is based on a very simple couple of formulaa. Even without formally knowing what a sequence is, it’s natural for them to put the fundamental angles in order, so I tried to see if a small formula could reduce the memory load.
Once defined the sequence of angles xn:
Then we have:
for n = 0, 1, 2, 3, 4.
Students tend to pick this up very quickly. It also reduces their anxiety when doing exercises, since instead of recalling a table, they just remember just 2 formulas and a straightforward index–angle association. If I explain it alongside a unit circle sketch, assigning n to each fundamental angle and then pointing out that signs just flip in the other quadrants, they start reasoning geometrically with less effort.
I’ve never seen this trick in textbooks. My guess is that it’s avoided because sequences haven’t been formally introduced yet, but textbooks often give formulas or notations before full explanations, just because they’re useful tools. At this level, a sequence is as natural as counting. At least in Italian textbooks, that’s the case. Is it the same where you are?