Through the Looking Glass:
Inverse Functions

UNH Mathematics Center

The inverse trigonometric functions ... judging just by their graphs ... are so odd-looking, that you might wonder just what they are good for.

It's a fair question. The inverse trig functions, especially the inverse sine, inverse tangent and inverse secant functions, turn out to be very handy as antiderivatives for a variety of integrals involving quotients and roots of polynomials. The reasons for this surprising development, in a nutshell:

• Polynomials with real coefficients can all be rewritten as products of linear, and irreducible quadratic, factors.
• The trigonometric functions satisfy the ''Pythagorean identity'' sin²(x) + cos²(x) = 1.

1  The sine function and its inverse

If you assembled all these ordered pairs, interpreted each pair as the coordinates of a point, and plotted the points, you would obtain the graph of the sine function.

1.1  The sine function is periodic, not one-to-one.

We have color-coded this sketch: the horizontal axis shows up on your screen, we hope, in red, and the vertical axis in blue. Each marked point's first coordinate is labeled in red and its second in blue.

The graph shows two periods of the sine function, with input numbers between -2p and 2p. The sine function's whole domain would be the entire set of real numbers; its range is the set of numbers (an interval) [-1,1]. The sine function is periodic: its graph is wonderfully repetitive. We've included four points whose second coordinate is Ö2/2. (A question for you: what is the first coordinate of the unmarked point on the graph?)

But the sine function's periodicity means that we cannot use the entire function when we construct its inverse - the inverse sine function, or the arcsin function. A function must always return a specific value for each input number: if x is a number and f a function, then f(x) must be the unambiguous name of one specific number.

1.2  Selecting a one-to-one portion of the sine function

The first thing we do is to select a portion of the sine function that is one-to-one (each input number is associated with exactly one output number, and vice versa). There would be lots of ways this could be done, and the graph below shows the an advantageous way to make the choice. The sketch to the left is the sine function's graph we saw just above: the sketch to the right is a portion of it that represents a one-to-one function.

Notice that values of the restricted sine function (the graph on the right) still range from -1 to 1, so that in a way we have kept ''as much of'' the function as we could, while making our graph show a one-to-one function instead of a periodic one.

There would have been many ways to have chosen just one section of the sine function's graph, in ''selecting'' a piece of the sine function to be one-to-one. Yes, there is a reason for making this particular choice: No, we are not going to burden you with it just now.

1.3  Making an inverse for the sine function

We make an inverse function out of original function's ordered pairs, by reversing each pair. The nice thing about a one-to-one function is that after we reverse the function's ordered pairs, the reverse pairs still satisfy the definition of ''function.''

A function is a collection of ordered pairs of numbers, such that no two (that is, two different) ordered pairs have the same first element.

A really good long look at the two graphs above - the restricted sine function's graph at the left, and its inverse function at the right - will show you the whole business in a nutshell.

• The first graph shows the ''restricted'' one-to-one portion of the sine function. We've marked the horizontal axis (where the input numbers live) in red. We've labeled three sample graph points, first coordinates in red. The output numbers (second coordinates), and their (vertical) axis, are in blue.
• On the right-hand graph you can see that we've made an inverse function by reversing the original function's ordered pairs. We've kept the same colors, so that now the it's the input numbers that are in blue, as is the horizontal axis where they live. The output numbers (the points' second coordinates) are in red, as is their (vertical) axis.
• Compare the two graphs! Every single point on one graph has had its coordinates reversed before being re-plotted on the other graph.
• One graph can easily be turned into the other. If we just got the red and blue axes in either picture repositioned as they should be in the other picture, all the rest of the graph would follow!
• You might think you could reposition the axes by rotating the sketch counterclockwise through one right angle, but it doesn't work that way. If you did that, the ''previously vertical'' axis would be horizontal all right ... but it would be pointing in the wrong direction.
• It will help you figure this out if you do it on paper. You can reposition the axes (and with them, the rest of the graph) this way:
  • Sketch the graph on paper. Pick up the paper, and turn it over. Hold it up to the light, and look at it from the back!
  • Now rotate the paper until you can see the ''old'' horizontal axis pointing upwards. The ''old'' vertical axis will now be horizontal, and its increasing direction will point to the right.
  • That's it! What can now see the same ordered pairs you had before, but their coordinates are reversed. You are looking at a graph of your function's inverse.

• The domains and ranges switch places. When we make an inverse for a one-to-one function, all the first coordinates of points from the original function become second coordinates of points on the inverse function's graph ... and vice-versa. That is, the original function's domain becomes the inverse function's range; and the original function's range becomes the inverse function's domain.

The function we end with is called the ''inverse sine'' function. It's also known as the ''arcsine'' function, and often enough (yes, we know the use of the -1 exponent is confusing) it's called ''sin^-1.''

The domain of the inverse sine function is the interval of real numbers [-1,1]. For any number x in that interval, arcsin(x) ... also known as sin^-1(x) ... is the number between -p/2 and p/2, whose sine value is x.

For instance:

• Many numbers have sine values of Ö2/2. For instance, sin(p/4) = Ö2/2, sin(3p/4) = Ö2/2, and sin(-5p/4) = Ö2/2. But sin^-1(Ö2/2) = p/4.
• Many numbers have sine values of -1. For instance, sin(3p/2) = -1, sin(-p/2) = -1, and sin(7p/2) = -1. But arcsin(-1) = -p/2.

2  An inverse for the cosine function

2.1  The cosine function is periodic ...

And so, just as with the sine function, it has no inverse function as it stands. Below is a sketch of two periods of the cosine function's graph.

There are any number of example sets of points on the graph, with identical second coordinates. What are the coordinates of the graph points labeled P, Q and R? Notice that these three points all have the same second coordinate.

2.2  A one-to-one portion of the cosine function

As before, we need to restrict the cosine function (so that the portion we ''keep'' is one-to-one) in order to form an inverse function from it. We choose to ''keep'' the portion of the graph for input numbers between 0 and p:

Our abbreviated graph represents a restricted cosine function. It's only a little piece of its former self, but it now has the advantage that it's a one-to-one function. You can see that it passes the ''horizontal line test'' ... No horizontal line meets this graph more than once.

2.3  Making an inverse cosine function

• First, provide new horizontal and vertical axes. Copy onto them (approximately) the same units as are on the restricted cosine graph's vertical and horizontal axes, respectively.

• Then, put a few points on your graph. You can use the marked points on the restricted cosine graph as landmarks. For each point, reverse its coordinates, and plot the reversed-coordinates-point on your graph.

• The domain of your inverse function should be the range of the cosine function ... which is the set of numbers between -1 and 1, inclusive.

• The range of your inverse function should be the domain of the restricted cosine function. What is that? (Hint: it shouldn't have any negative numbers in it.)

• Now you can ''fill in'' the rest of your inverse function's graph. Make up more landmark points if you like, or copy the restricted cosine function's graph onto paper, turn it over, and hold it up to the light! (If you prefer, you can imagine rotating it around the line y = x: it amounts to the same thing.)

Ready? You should have a picture like the graph below.

This is a graph of the arccos or inverse cosine function, often written with the -1 exponent as cos^-1. Its domain is [-1,1], the same as the range of the cosine function. Its range is [-p/2,p/2], which is part of the cosine function's domain. (That is, all of the restricted cosine function's domain.)

For example, arccos(Ö2/2) = p/4 because

• cos(p/4) = Ö2/2 and
• p/4 is between 0 and p ... that is, p/4 is in the range of the arccosine function.

2.4  The inverse tangent function

We'll just describe quickly how the inverse tangent function is constructed. After following the description of the inverse sine and cosine function, we think you can follow the construction of the inverse tangent easily. It's probably in your textbook just waiting for you. Just a few outline points:

• The tangent function is periodic, with period p. It has vertical asymptotes p units apart, and its range is the entire set of real numbers.
• The ''selected portion'' of the tangent function's graph, for the purposes of constructing an inverse function, is the portion between the vertical asymptotes at x = -p/2 and x = p/2. If you delete all the rest of the tangent graph, you can see that this remaining section of it is a one-to-one function.
• You might select landmark points (-p/3,-Ö3), (-p/4,-1), (0,0), (p/4,1), and (p/3,Ö3).

• When you reverse all these points remember that the asymptotes reverse too. Everything vertical becomes horizontal! Your inverse function will have horizontal asymptotes.
• The inverse tangent function's domain is the whole set of real numbers. The range is an interval of real numbers. (Does the range-interval include its end points, or not? Think about those horizontal asymptotes!)

• You can find a picture of the inverse tangent function's graph (''arctan'') in your text. Probably on your calculator too! ... where the function is probably called tan^-1, to save space. Does it look like your sketch?
• What (approximately) is arctan(75,000)? Where is it on your graph? What (approximately) is arctan(-0.02)? Is it positive or negative? Where is it on your graph? Can you give - without looking it up - an approximate value for arctan(-1.15)?