More inverse trig functions, and a few others

Having looked at the inverse of the tangent function, we’re now going to look at the other inverse trig functions. Let’s begin with the case of the sine function. As we know, that’s what its graph looks like. Now we need to restrict the domain so that on that restricted domain the sine function is injective and also attains all its relevant values. The obvious choice here, and it’s the choice that is always made, is the interval minus pi over 2 to pi over 2. We see that on that interval, the function is strictly increasing. And it takes all the values between minus 1 and plus 1. So we consider that function with that domain and that interval. It’s then a bijective function.

Therefore, we can define an inverse function. And the name we give to that inverse function is arcsine, also sometimes denoted inverse sine. So it’s a function that maps the other way. It maps the interval minus 1, 1 back to the interval minus pi over 2, pi over 2. To summarize then, we’ve looked at the sine function between minus pi over 2 and pi over 2, restricted domain. And we’ve constructed the inverse function in the following way. We’ve said that for each t in minus 1, 1, there is a unique x in that interval for which t is equal sine x. There’s the t, and there’s the x.

And we’re defining arcsine of t to equal that value of x, also called inverse sine function and denoted this way. What does the graph of arcsine look like? Rather easy to find. You apply the usual– one of the two usual methods we have geometrically to find the graph. You get this. As you can see, arcsine is defined on the interval minus 1, plus 1, and not otherwise. What about the inverse of cosine. Well, it’s going to be very similar. One difference though is that you can’t take the interval minus pi over 2 to pi over 2, because on that interval, the cosine is not injective, as you can see. So you shift the interval.

You take the interval 0, pi, instead for inverting cosine. On 0, pi, you see that cosine is strictly decreasing, and hence injective. So you obtain an inverse function. It’s labelled arccosine. And it maps minus 1, 1 to the interval 0, pi. Easy enough to find what the graph looks like. It turns out to be like this. Notice that arccosine, also called inverse cosine, is defined just on the interval minus 1 to plus 1. There are a few forgotten functions that we’re not going to discuss in any detail. As you know, our discussion of trigonometry, which is now drawing to a close, has defined three functions, sine, cosine, and tangent, as well as their inverses.

But depending on what continent you live on, you may also run into other trigonometric functions that are current in trigonometry, for example, cotangent. Cotangent x means the cosine over the sine, equivalently 1 over the tangent. The secant of x is 1 over the cosine. And the cosecant is 1 over the sine. And these functions also have inverses that can be defined in similar ways to what we’ve done. And if you’re dealing with all of these functions, then there are of course, even more identities that you might have to know. For example, it’s easy to prove that tangent squared plus 1 is equal to secant squared.

However, it is possible to do everything you want in trigonometry without dealing with these extra functions, as some countries do. Another class of functions we have not discussed is related to the exponential function, e to the x. These are the so-called hyperbolic functions. They were defined and introduced in the 19th century. They’re used a lot by engineers, for example, because certain differential equations involve these as solutions. What are these functions? Well, here they are. They’re read usually as hyperbolic sine, hyperbolic cosine, and hyperbolic tangent. And as you see, they’re completely defined in terms of the exponential function. And they also admit inverses, which have some uses. And they have a whole bunch of identities of their own.

Actually, they’re rather amusing in the sense that the identities for these hyperbolic functions look a lot like trigonometric identities with occasionally a sign being different. There’s a circular identity, for example, that says that hyperbolic cosine squared minus hyperbolic sine squared is equal to 1. Now, there are so many functions and so little time that we’re not going to do them all. But I can assure you that we have done the ones that count the most.