The function arctan

Do you remember how we used the inverse function concept to define logarithms? We started with the exponential functions. We took their inverse functions. And we got logarithms. It turns out that, for trigonometric functions, inverses are important, too. They do, however, present certain difficulties that we will now discuss and show how to deal with. Consider an equation such as sine x equals y. This will either have no solutions at all, or else it’ll have an infinite number of solutions. For example, there is no x such that sine x is equal 2, because sine is always between minus 1 and plus 1.

But if you do have a solution to sine x equals y, then you’ll have an infinite number, because the function sine is periodic. And so you’ll have lots of other solutions. Graphically, this is very clear. If I give you a y, for instance, that is strictly greater than 1, there are no points x for which sine x will equal that y. Whereas, if I give you a y between 0 and 1, then there are many points, x, whereas sine x will equal that y. The function sine, in other words, as a mapping from R to R, is neither surjective– it’s not onto – nor injective. It’s not 1 to 1.

The function tangent is a little better at first glance, because the equation tangent x equals c will always have an infinite number of solutions x. That follows from the nature of the graph of the tangent function, which we know. If I give you any value of y, it’s clear that there are many points where the horizontal line corresponding to y cuts the graph of the tangent function, so many points x for which tangent x is equal y. In other words, the tangent function is surjective. But it’s not injective. Now we know how to cure this lack of injectivity. It’s a matter of restricting the domain of the functions.

We see that one of the solutions to tangent x equals y will lie in the open interval minus pi over 2 to pi over 2. For that y, there will be such an x. And we will single that one out. That is, we will observe that the function tangent as a function restricted to the interval minus pi over 2 pi over 2 is both surjective and injective. Therefore, it admits an inverse. Those are the properties we need to define an inverse function. And we define the inverse that way. Now the inverse to tan viewed this way is called arctan. It is also sometimes called inverse tan, and denoted tan to the minus 1.

So arctan of y means the unique x in the open interval minus pi over 2 to pi over 2, which satisfies tangent x equals y. There’s a y. There’s the corresponding x. Arctan as a function from R to that open interval– a function and goes back the other way– is surjective and injective as an inverse function. What is its graph going to look like? Well, we know how to construct the graph of an inverse function from that of the original function. Here’s the original function, tangent, between minus pi over 2 and pi over 2. We know that we want to reverse the role of vertical and horizontal. So what we do is we perform a rotation to do that.

But we want the vertical axis to be pointing upwards. So we do a flip. And now we have the graph of the function arctan. A few facts about this function, the function arctan is 0 at 0.

The arctan of 1 is equal pi over 4. Why? Because pi over 4 is the unique angle between minus pi over 2 and plus pi over 2, whose tangent is equal to 1. Arctan is an odd function.

Tangent of arctangent t will always be t. The two functions cancel one in one another out in this sense. But careful– is it always true that the arctangent of tangent t is t? No, only if t is between minus pi over 2 and pi over 2. Because that’s the range of arctangent. Otherwise, the graph shows that you’ll have two horizontal asymptotes. As x approaches plus infinity on the right, the graph will approach the horizontal line y equals pi over 2. And contrariwise, as you move far to the left towards minus infinity, the graph will approach the value minus pi over 2.

We had another way, of course, of generating the graph of the inverse function from the graph of the original function. You may recall that it involved reflecting the graph about the line y equals x. If we do that in this case, we see that we get, in fact, the graph of the function arctan.