Cyclometric arcus function

For the basic goniometric functions to be meaningful, their inverse functions also need to be defined. These are called arcus functions, more precisely arcus sine, arcus cosine, arcus tangent and arcus cotangent. Also abbreviated as arcsin, arccos, arctan and arccot.

Motivation

Let's start with an example. Calculate the magnitude of the angle alpha:

We know the lengths of two sides in a triangle, side b = 3 and side a = 6. We have to calculate the angle alpha - the side a is the hypotenuse and the side b is the hypotenuse opposite to the angle alpha. The sine function works with the hypotenuse, more precisely the sine of the angle alpha is equal to the ratio of the length of the hypotenuse to the length of the hypotenuse. So let's try to plug that into the formula:

$$\sin(\alpha)=\frac{|b|}{|a|}=\frac{3}{6}=\frac12$$

We now know that the sine of the angle alpha is equal to one half. The question is how to get the angle from knowing this number. To do this, we need the inverse function to the sine function. The sine function takes an angle as input and returns the ratio of the two sides. But now we need a function to put the ratio into and get the angle.

This is exactly what the arcus function does. Before we properly define what arc functions actually are, let's calculate an example using Google that tells us that the arcsin of one half is 30 degrees. If you wanted to calculate the same example on a calculator, that's where arcsine is often labeled as sin⁻¹.

Inverse function to the arcsine?

First, look at the graph of the sine function:

Now let's review some properties about functions. The function f has an inverse function if it is simple. What does this mean? That if we take two elements x₁ and x₁ from the definitional domain, then it must be true that their images f(x₁) and f(x₂) are different. This must be true for all possible pairs. It is easy to see from the graph - if we can intersect the graph with a line that is parallel to the axis of x so that this line intersects the graph of the function f more than once, then the function is not simple.

Obviously, for the sine function we are able to find such a line. For example, the x axis itself intersects the graph of the sine function more than once, or more precisely, it intersects it infinitely many times. Therefore, the sine function is not simple and so there is no inverse function to it.

Inverse function to restrict the sine function

We know from the last chapter that there is no inverse function to the sine function. However, we know from the first chapter that there is a function arcsin that behaves exactly as we want. How did we achieve this? We selected only the part of the sine function that is simple, so we can define an inverse function to this restriction of the function.

We achieve the restriction of the sine function by shrinking its defining domain. What subset of the original defining domain(real numbers) should we choose? We can choose several subsets, but the best choice is the interval

$$\left<-\frac{\pi}{2}, \frac{\pi}{2}\right>$$

This interval is shown in the following figure:

We are already able to define the inverse function on this defining domain because the sine function is simple on this interval. The inverse function is symmetric along the axis of the first and third quadrants with the original function, so we can already plot the graph of the function that will be the inverse of our restriction function sin(x). See the figure:

The original sine is plotted in blue. The part of the function we are looking for the inverse of is highlighted in red. In green is the inverse of the function we label arcsin or just asin.

The definitional domain and the domain of values of arcsin

We can use these two functions to nicely illustrate the swapping of the definitional domain and the domain of values. Take a look at the following image, it has the same graphs, in the same colors, just enlarged a bit and some lines and points added:

Notice that the definitional domain of the restriction of the sine function (red line) was the interval

$$\left<-\frac{\pi}{2}, \frac{\pi}{2}\right>$$

By definitional domain we mean all x, for which the function is defined. We just defined it for this interval to get a simple function. On the graph, we look for these x on the x axis.

Now see what the range of values of the function arcsin looks like (green line). The range of values are all the y that the function can return to us in the output, so we're looking for them on the y axis. The arcsin function's range of values is again the interval

$$\left<-\frac{\pi}{2}, \frac{\pi}{2}\right>$$

this is a general property of the inverse function, so it shouldn't surprise anyone. Similarly, this is true for the range of values of the restriction of the sine function and for the definition range of arcsin. The range of values of the sine function is the interval from minus one to one. Sine can only return us these values, the sine of any angle is not greater than one or less than one. And since the sine of no angle can be greater than one, it stands to reason that the inverse function also cannot take an input value greater than one. Therefore, the defining domain of the arcsin function is also the interval <−1,1>.

You can calculate the value of the arcsine function in the calculator here.

Arcus cosine

Now just briefly on the inverse function to cosine. For the same reason as for the sine function, there is no inverse function here, because the function is not simple. However, we can select a subset of the defining domain to obtain a simple function.

The blue part is the graph of the cosine function, and the red part is the restriction of this function to the definition domain <0,π>. The green line then represents the graph of the function arccos(x), the inverse function to our restriction of the cosine function.

Arcus tangens

The graph of the tangent function looks like this:

The graph has a different shape than the previous function, but is still not simple. But again, we can do a restriction of this function and choose a part that is simple. We choose an interval

$$\left<-\frac{\pi}{2}, \frac{\pi}{2}\right>$$

The inverse of the function then has the following shape:

Arcus cotangens

All the same as in the previous paragraphs. The graph of the function cotangens: The defining domain of the restriction of the function cotangens will be <0,π>.