<p>I memorise after I’ve resolved what’s going on. After that you find it’s very easy to memorise because you know exactly why each element is there. </p>
<p>For identities, I was referring to the proofs once the basic derivative relationships between sin and cos are established. The idea is actually pretty to grasp – during my precal tests I always used my school clock, neatly divided into 12 hours (segments of pi/3!). In precal I naturally noticed that the sin increased rather quickly then slowed down from 3 o’clock (0 radians) until 12 o’clock (pi/2); and that cos decreased rather slowly than sped up from 0 until pi/2. </p>
<p>Then later I was taught d[sinx]/dx = cos x, and that d/dx cos x = -sin x, etc. and for a while I accepted this, not really understanding what that relationship entailed (although it was pretty cool that the circular functions had … circular derivatives that cycled on each other). Then later I was reading an ancient physics text that had a table gave cos, sin and tan values for each 0.01 of a radian up to pi/2 radians. </p>
<p>It wasn’t long before I remarked, “gee, the difference between each of the sin values seems to be equivalent to the cos value beside it … a lot of this table seems redundant, even for an era without calculators.” Then I glanced down and saw that the magnitude of the rate of change [change in y-value per radian] for cos was equivalent to the sin value beside it. It was only THEN did the full significance of the idea of the derivatives of cos and sin all circularly leading to each other hit me, because how convenient is it to have two sets (of four, unless just uses the absolute value) of values whose rates of change in one list are based on the values in another, and that list’s rate of change being based on the values of the first? They’re like circularly defining themselves, almost like a differential equation on a different level almost, because their rates of change are based on each other.</p>
<p>Of course, this property is probably why sin and cosine are intricately connected to e, because of this property of recursively defining themselves (I mean, when one is trying to view the relationship fundamentally, and I’m not even talking about Taylor series and Euler’s formula and whatnot).</p>
<p>I mean, could you create a various tables of values whose rates of change were all based on the table next, and the next, but eventually feeding back into the first one in a loop, that was not sin and cos and their respective opposites? (e^x doesn’t count because except for itself it doesn’t really loop once x has a coefficient more than 1)?</p>
<p>I mean, this is the stuff that I wouldn’t have gotten with the formula until I thought it out for myself. I mean, I had been using “look at the clock to estimate or verify trigonometric values or trends” strategy for years but I had never put 2 and 2 together when I was first told the trig derivative formulas.</p>
<p>It’s dangerous to give a formula without investigating the implications of the relationships outlined by the formula. And if you learn conceptually rather than by rote memorisation of the formulas, well you find come test day you have internalised the relationships already. </p>
<p>I just don’t mean knowing how to derive the formula. You can derive the formula without knowing what the implications of the relationships are. Visualisation of what is happening as you apply the formula is key.</p>