[AM_20151119] Anticipation and Bridging as Proof Tactics

Question

Introduction

     In a previous article, I have showed an “evil” tactic that can be used against “evil” questions.  Here is another “evil” trigonometric proof question.

     Many traditionalist school teachers insist on starting either from the LHS or the RHS, and working all the way to the other side.  Personally I do not mind any form of presentation as long as it is logical.  But not many students are able to do that.  People tend to fall into the trap of beginning a proof with the statement that they are supposed to prove in the first place.  This is called circular reasoning (or “begging the question” or petitio principii).  It is definitely a no-no.  So there is some advantage to sticking to the traditionalist mould.  The disadvantage is, of course, that it stifles creativity and this gives a misleading image of mathematics to the learner.

More “Evil” Tactics

     Now, if we do not want to “break the rules”, perhaps we can “bend the rules” a little.  On a piece of rough paper, or in your mind, secretly work from both sides and try to bridge them in the middle.  Let us compare

Think:  How are they similar?  How are they different?

As you can see, the LHS already has a preponderance of  cos 75°.  One of these  cos 75°  must somehow disappear.  The  RHS  has  4 sin 75°  which the LHS does not have.  So if we start from the LHS, we can use our magic wand [SV4] to create  4 sin 75°  out of thin air, remembering to divide by the same thing, so that the original value does not get changed.

Solution

Reflection

     Have you learned anything from this problem?  Let us review the heuristics used to help us solve this problem successfully.

Heuristics Used

H04. Look for pattern(s)

H05. Work backwards

H09. Restate the problem in another way

H10. Simplify the problem

H11. Solve part of the problem

Special Variants of Heuristics (Good for Trigonometric Proofs)

SV1   Work from both sides and try to bridge them in the middle [~ H04]

SV2   comparing (similarities/differences) what you have now with what you want [~ H05]

SV3   anticipate what will happen in the end and what you must do now [~ H05]

SV4   Magic Wand or create something out of nothing (无中生有) [~ H09]

Suitable Levels

* GCE ‘O’ Level Additional Mathematics

* IB SL & HL Mathematics (revision)

* other syllabuses that involve trigonometry

* just about anyone who is interested, really