monkey3 wrote:What do you learn after this if your aim is learning calculus ?
I'm not sure what you mean by "learn". You posted a great list of trigonometric identities, which are useful tools in actually doing
trigonometry. But it almost seems as if you equate memorizing these tables with "learning trig". For example, missing on this page is the unit circle, which visually conveys what "sin" and "cos" etc. mean
In any case, trig is not a precursor to calculus. It's just that a lot of the useful applications of calculus require trig anyway, and trig gives you a whole bunch of new functions to play calculus with, so it's good to have those tools under your belt first.
Calculus is first and foremost about rates
of change. Things change over time (or distance, or whatever), and at any given (one) point there is a certain rate of change, but you can only "see" the change by comparing two points. So, you consider two points "very" close together and get an approximation (because it's changing!)... the closer together your two points the better the approximation. Calculus is about taking this to eleven
For that, you need the idea of a limit. As "delta x" (the difference between the two points) goes to zero, the value of the expression you are looking for will get closer and closer to some value; that value is the limit.
Trivial case: y=2x. As x approaches 4, y will approach 8.
Problematic case: y=1/x. As x approaches 0, y increases without bound ("approaches infinity").
Useful illustration: y=x2
As x approaches 0, what happens? The bottom makes the expression want to blow up ("approach infinity"), but the top looks well behaved. You can't just divide both sides by x to simplify, because at x=0 you're dividing by 0. However, you can do that as long as x doesn't
equal zero. What happens? We simply get y=x in those cases. As x approaches zero, y also approaches zero. The limit
of the expression x2
/x as x approaches 0 is
More interesting expressions where the limit isn't obvious are available in a zillion textbooks. But that's the idea of a limit.
We then apply that idea to rates of change, to find (say) the slope of a curve. That curve needs to be expressed first in math, for example,
Slope is rise over run; that's easy if you have a straight line. Just pick two points and do arithmetic; the slope is constant. But with this curve, the slope keeps changing. We want an expression for the slope of this curve at any point. To do this, we still pick two points. One of them will be x (the point at which we want the value of the slope), and the other will be nearby: (x+a), where a is small.
Slope is rise over run. The run is easy... it's the horizontal distance between the two points It's just a.
Algebraically, it's (x+a)-x
(the x coordinate of the second point, minus the x coordinate of the first point)
The rise is a little harder; we need to use the formula for our original curve and take the difference of the two resulting values. For the first point it's just
For the second point it's
Subtract the two, and the rise is:
rise = 3(x+a)2
rise = (3x2 + 6xa + 3a2)
rise = 6xa + 3a2
so the slope = rise/run = (6xa + 3a2
)/a = 6xa/a + 3a2
Now, take the limit as a=>0
(as the distance between the two points we used to calculate the slope approximation vanishes)
We get slope = 6x
The a in the denominator can never be
zero, but we can see from the expression that the closer we get to a=0, the closer the equation comes to y=6x.
The slope actually increases (linearly in x). As x moves to the right, the slope gets steeper. (and as it moves to the left of zero, it gets steeper in the other direction). When x is zero, the slope is zero, and the curve is at a minimum (or maximum).That
is calculus in action.
No trig involved. But of course trig functions have slopes, and there's lots of fun to be had with calculus on trig functions.