## Monday, July 1, 2013

### Unizor - Trigonometry - ARC... Introduction

We have introduced six trigonometric functions in the previous lectures: y=sin(x), y=cos(x), y=tan(x), y=cot(x), y=sec(x) and y=csc(x). Each is a function of an angle (usually expressed in radians) chosen from a domain of real numbers with a range (also a real number) depending on a specific function.
In this lecture we would like to address the issue of inverse functions for these six trigonometric functions. Inverse function, in general, is a function with a domain being a range of an original function, with a range being a domain of an original function and a nice property that, applied in composition with an original function, it results in an identity function either over a domain or a range of an original function, depending on the order of composition.

From this perspective an inverse of any trigonometric function is a function defined on some subset of real numbers (that is, on a range of an original trigonometric function) and values among the angles, which are a domain of an original trigonometric function (usually, expressed in radians).

The main tool to define trigonometric function was a unit circle. In this circle an angle in radians corresponds to the length of an arc that corresponds to this angle. That is why all inverse trigonometric functions are called by the name that combines a prefix arc (implying that we are seeking an arc or a corresponding angle in a unit circle) and the original name of a trigonometric function. Thus, we can talk about y=arcsin(x) as an angle, a sine of which equals to x, or about y=arccos(x) as an angle, a cosine of which equals to x, etc.

There is a problem, however, which makes this definition plainly wrong. All trigonometric functions are periodic and, therefore, there are multiple (actually, infinite) number of angles with the same value of any trigonometric function. For example, sin(π/6)=1/2, sin(5π/6)=1/2, sin(π/6+2π)=1/2 etc. So, what is arcsin(1/2)? It can be π/6, it can be 5π/6, π/6+2π etc. - an infinite number of values. This cannot be a definition of a function that for each argument has to be fully and uniquely defined with one value. To overcome this difficulty in defining inverse trigonometric functions we have to re-examine a concept of an inverse function, which is a subject of this lecture.

Recall that a function is a rule that puts into a correspondence (or "maps", as sometimes say) each element of one set (called domain) to a single element of another set (called codomain) with all the elements of a codomain that actually have a prototype in a domain (that is, actually are mapped into) called a range of a function. So, a function is a triplet of a domain, a codomain and a rule that transforms each element of a domain into some (single) element of a codomain. It is important to understand that we can change the definition of a function by just changing a domain or a codomain without changing the rule of transformation of elements of one into elements of another. Range specification is not part of a function definition, it's the consequence of it.
It's important to emphasize that the rule of transformation must be applicable to each element of a domain with no exceptions and the rule must map each element of a domain into one and only one element of a codomain. Also important to understand that a range does not necessarily coincide with a codomain, it can be its subset.
Notice that it is possible for two different elements of a domain to be mapped into the same element of a codomain, thus a value of a function for two different arguments can be the same.