**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 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.

*petitio principii*). It is definitely a no-no.**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.

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

## No comments:

## Post a Comment