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.
Now let's address inverse functions.
In short, we would like to know an argument of a function (an element of its domain) if we know the value from a codomain this argument is mapped into.
Unfortunately, it's not always easy. It is easy in case a function establishes one-to-one correspondence between its domain and its range that completely fills the codomain. For example, an exponential function y=10x is such a function. It maps a domain of all real numbers (an argument x can be any real number) to a codomain of all positive real numbers (a value y=10x is always a positive real number) with its range completely filling a codomain and, importantly, any two different arguments x1 and x2 (x1≠x2) produce different results y1=10x1 and y2=10x2 (y1≠y2), thus establishing one-to-one correspondence between a domain of all real numbers and a codomain of all positive numbers. Therefore, we can easily find an argument if we know the value of a function, thus mapping all positive real numbers into all real numbers. This mapping is a definition of a logarithmic function y=log10(x) which is inverse to an exponential function y=10x with its domain being a codomain of an exponential function and its codomain being a domain of an exponential functions.
Let's consider another familiar example, y=x2 defined for all real arguments. It perfectly corresponds to a definition of a function mapping all real numbers to all non-negative real numbers. However, it's not a one-to-one correspondence. Numbers 2 and −2 are both mapped to a number 4. Generally, two numbers A and −A, where A is any real number, are mapped to the same value A2. How can we find an argument in this case if we know the value of a function? Strictly speaking, without any additional conditions, we cannot. Defined as above, the function y=x2 has no inverse one.
There is however a way to define a function that is almost what we need. Recall that monotonic (or monotonous) function of real argument with real value always defines a one-to-one correspondence between its domain and its range. Let's narrow a definition of a function y=x2 to an interval where it is monotonic. We have two such intervals: (−∞,0] and [0,+∞). On the former the function is monotonously decreasing, on the latter it is monotonously increasing. Let's concentrate on the latter, that is x≥0 and use only this interval as a domain. Since on this interval function y=x2 is monotonously increasing and still fills completely the range of an original function y=x2 defined for all real arguments, the narrowly defined function establishes one-to-one correspondence between all non-negative arguments and all non-negative values of a function. Therefore, an inverse function exists and can be defined. Its domain will be a set of all non-negative real values and the same is true for its codomain and a range. Thus, the value for such inverse function for an argument 4 will be 2 (and not −2, since −2 does not belong to its codomain, regardless of the fact that (−2)2=4). This restriction for the codomain of our newly defined function, which, incidentally, is called principal square root, allows to define a function inverse to a narrowly define function y=x2. More precisely, if we narrow the domain of a function y=x2 to non-negative arguments x, the principal square root function y=√x represents an inverse function to it.
The above technique of restricting the domain of an original function in order to be able to define an inverse function is used to define inverse trigonometric functions. If we narrow the domain of a trigonometric function to an interval where there is a one-to-one correspondence between thus restricted domain and the full range of a function, the inverse function can be defined. If this interval exists, we can consider an original trigonometric function only on this interval and define an inverse trigonometric function as having codomain coinciding with this interval.
Monday, July 1, 2013
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