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Extracts from chapters : 1 | 2 | 8 | 12
Lesson 12
 
Equations are good, but inverse functions are better!
The function inverse to cosine
Analysis of the function as inverse to y = x2
The rule for construction of the graph of an inverse function
Role of monotonic regions
Definition of the function y = arccos x as an inverse to cosine
Graph of the function y = arccos x
Solving the equation cos x = A for any A

 
  Lesson 12

 

 

 

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?…

 
 

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