| Dad: |
So,
guys, we have the problem: how to solve
the simplest trigonometric equations
sin x = A and cos x = A
in general form, i.e. for any given number
A from the interval [–1, 1]. |
| Nick: |
I have
a feeling that it isn’t so simple. |
| Dad: |
It depends. |
| Nick: |
What
do you mean, “it depends”? It
is either simple or it’s not. One of
the two! |
| Dad: |
It depends
on what we mean by the term “solve”. |
| Nick: |
Isn’t
it obvious? To solve an equation means to
find all such numbers x, that if they are
substituted into the equation, we will get
an identity. |
| Dad: |
That’s
true, but what do you mean by the word “find”? |
| Nick:
|
“Find”
– means to indicate which x must be
taken. |
| Dad: |
Indicate
in what form? Since, in a general case, x
will most likely be some irrational number. |
| Nick: |
Well,
we must represent x in such a form that would
allow to calculate it to any degree of precession. |
| Dad: |
I agree,
this would be the perfect solution, but we
cannot reach this solution right now. |
| Nick: |
So in
what form will we solve these equations? |
| Dad: |
In such
a form, that we simply denote their solutions
with some symbols. |
| Nick: |
And that’s
all?! |
| D: |
Yes,
you can say that. |
| N: |
And after
this, you claim that we will solve something?!
In my opinion, it is a scam! I can also tell
the whole world that I am able to solve any
equation. Bring it on! I will quickly invent
some name, for example “big bird”,
and will tell everybody: “The solution
is found! It is big bird!” But it’s
obvious that I found nothing. Therefore, designations
alone give us nothing. |
| D: |
Not
at all. Let’s, for example, solve
the equation:
s2= A ( A >
0)
with respect to unknown s. |
| N: |
What
a problem! Obviously, s =  .
Oh no! More exactly, s = ± . |
| D: |
And that’s
all? This is your final result? |
| N: |
Yes!
What else? |
| D: |
But where
is the method of finding s? |
| N: |
Hmm…
Probably the method is inside the notation
. |
| D: |
Exactly! We
used notation
in order to solve the equation. And we can
stop at that, saying that all is done. How
to calculate
is a separate problem. The same things work
in solving the simplest trigonometric equations. |
| N: |
It’s
amazing! I would never believe in my whole
life that something could be reached with
only notations. I used to think that coming
up with just a name is not a big deal. |
| D: |
It might
seem so at the first glance, but in reality,
introduction of a proper notation is a big
accomplishment. It is as though we “materialize”
or create an object that didn’t previously
exist in our consciousness. As a result, the
base for its study appears. By the way, recall
that we began the study of trigonometry exactly
with the introduction of notations for trigonometric
functions. After that we discovered many properties,
which allowed us to calculate their values
for certain angles. In truth, we didn’t
get a general method for approximate calculation
of trigonometric functions, but we found out
many useful properties. The development of
approximation methods is a separate task that
is resolved in other disciplines. |
| N: |
Then,
we don’t need to do anything with our
simplest equations? Somehow we’ll denote
their solutions and say good-bye? |
| D: |
Yes,
in principle, but there are a few subtle points
which we need to carefully examine. |
| N: |
What
subtle points?…
|
as inverse to y = x2