{∅, {∅}, {∅, {∅}}} : Rage Against the Meat Grinder

Since our local search engine is, let's face it, close to useless, I could not easily find the thread about slow thinking and repetitive learning styles, but this is as good a place as any to document an example from lived experience.

Suppose I am "working" through the exercises in a text I consider to be "elementary" just as a review, and I come to a problem that involves some thinking about trigonometric identities.  Now, let's assume I have a stack of textbooks I want to "get to," and one on top, say a Linear Algebra text, that I am most committed to, almost feeling as though I am just goofing off by working through Joyner's Differential Calculus and Sage.   I come to a problem where I have to prove that the limit (as h approaches pi/2) of cos(theta - h)/cos(2*theta - h) is -tan(theta).

Sage computes result as sin(theta)/sin(2*theta)

There is the temptation to skip this pesky little problem since I want to "get to" something else.

Aha, and this is an example of what I am referring to when I plead the case for SLOW THINKING in the spirit of bringing some "fun" into the learning process. 

This would involve what I am calling a mind shift.

This is the benefit of learning outside the confines of an academic or corporate setting: to have the inner (intellectual) freedom to respect one's own lack of clarity enough to CARE about the what and the why and the wherefore. 

[enter demon, Subscript One]: Now, it's getting late in the game, Herr Hallar ... do you have time to understand things in such a thorough manner?

[enter second demon, Subscript Two]: I'm afraid I'm going to have to INSIST we move slowly through this material, sir!  Do not piisss on trigonometric relations while calculating limits in an effort to go racing up the metaphorical mountain so as to get down to the serious business of trying to understand what differential equations mean!  Are you afraid you shall be suddenly ripped from your chamber and sent into the abyss before reaching your intended destination? 

[Subscript One]: There's no time to goof off and actually understand!!!! 

[Subscript Two]: No time?

[Steppenwolf]: I must face that which perplexes me with courage and show my disdain for those who act is if everything is trivial.  I have to agree with Subscript Two.  We can afford to go on a tangent if it promises a mind shifting experience.



[enter third demon, Subscript Three]:  Hint:  sin(theta) == cos(2*theta), as in sin(pi/6) == cos(pi/3) = 1/2 

second hint:  tan(-theta) = -tan(theta)

tan(-pi/6) = -tan(pi/6)
_______________________________
cos(pi/6 - pi/2)/cos(pi/3 - pi/2) == cos(-pi/3)/cos(-pi/6) = -tan(pi/6)

A few more tid-bits:

cos(-2*theta) = cos(2*theta) = sin(theta)

cos(-theta) - cos(theta) = sin(2*theta)

tan(-theta) = -tan(theta)
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Where I get jammed up is that sin(theta)/sin(2(theta) = tan(theta), not -tan(theta)

Note that cos(theta - pi/2) = cos(2*theta) = sin(theta)

Also, cos(2*theta - pi/2) = cos(-theta) = cos(theta)

so, as limit of h approaches pi/2,
cos(theta - h)/cos(2*theta - h) approaches cos(2*theta)/cos(theta) = sin(theta)/sin(2*theta) = tan(theta), not -tan(theta) !!!!

- and Sage agrees!   Those who publish textbooks sometimes make mistakes.

Another CAS computes result of limit as 1/(2*cos(theta)) which is the same.

Check:  1/(2*cos(pi/6)) = 1/(2 * (sqrt(3)/2)) = 1/sqrt(3) = sqrt(3)/3

And after spending the entire morning on "one easy little exercise," while my understanding isn't exactly intuitive, one thing is clearer to me, and that is the definite relationships existing between the trigonometric identities.

Do you think that the authors of such texts have that kind of time to consciously lead the attentive reader into specific conceptual arenas?  They must exist in eternity.  I mean, there has got to be some kind of transcendental realm where "time" no longer presses down on us.  That's the "place" where we can actually enjoy the thinking process and momentarily transcend drudgery.

Maybe ...