Inverse trigonometric functions (Lecture + Video)

Inverse trigonometric functions Lecture

Topic: Inverse trigonometric functions


During the lecture I will introduce the concept of cyclometric functions: arcsinx (in english notation: {sin}^{-1} {x} ), arccosx (in english notation: {cos}^{-1} {x} ), arctgx (in english notation: {tan}^{-1} {x} ), arcctgx (in english notation: {cot}^{-1} {x} ). These are inverse functions to trigonometric functions.

Let’s stress once again, that in below lecture notation is:

arcsinx is equivalent to {sin}^{-1} {x}

arccosx is equivalent to {cos}^{-1} {x}

arctgx is equivalent to {tan}^{-1} {x}

arcctgx is equivalent to {cot}^{-1} {x}

The lecture consists of two parts. In the first one, I only show how to quickly calculate the values ​​of inverse trigonometric functions, without going too deep into the topic (this part is accompanied by a video, a fragment of my Course on Definite Integrals and Applications of Integrals (in polish) ).

In the second part, I describe inverse trigonometric functions more precisely, show their graphs, etc.

To understand the lecture you will need:

  • trigonometric functions (high school)

Part I

Inverse trigonometric functions – “INSTANT” version

Inverse trigonometric functions “in common sense” are simply the opposite of trigonometric functions. So arcsinx (or {sin}^{-1} {x}) is the inverse function of sinx.

That is, if, for example, we know that , it means that .

And so on:

In addition, we have a few properties of inverse trigonometric functions that allow us to calculate their values ​​also for negative arguments:

So we can also calculate this:

So, if we have a table of values if trigonometric functions, we can easily determine the values ​​of inverse trigonometric functions from it, simply by reading it “the other way around”.

I explain it in more detail here in the video:

Table of basic values ​​of trigonometric functions from the video – download here .

Part II

Inverse trigonometric functions – full version

Introduction – why part one is not enough

So it looks like in Part One we defined each inverse trigonometric function as the inverse of its corresponding trigonometric function.

Let’s formalize this a bit. We said, for example, a function takes a value when function from this is equal .


That is, if we want to calculate we wonder what the cosine of the angle gives , we realize that it is an angle and we have the result: .

Does this exhaust the topic of the values ​​of inverse trigonometric functions?

Of course NO .

Let’s look at the whole reasoning again using specific numbers:

If we want to calculate we wonder what the sine of the angle gives , we realize that it is an angle and we have the result: .

Where’s the problem? In the bolded part:

If we want to calculate we wonder what the sine of the angle gives , we realize that it is an angle and we have the result: .

Unfortunately, not only sine is equal to .

Let’s recall the graph of the sinx function (I marked the value on it):

Sinx chart with the value 1/2 marked

You can see and we already know it from high school that the sine reaches a value not just for the angle , but also for angles:

That is

So let’s recall once again our way of calculating arcsin:

If we want to calculate we wonder what the sine of the angle gives , we realize that it is an angle and we have the result: .

Well, now we know that it’s not just sin gives , so it looks like:

This would mean actually, that arcsinx ({sin}^{-1} {x}) is not a function at all, because one argument has several values ​​assigned to it!

Giving a clear answer to the question of what the arcsin ({sin}^{-1}) of something is equal exactly would then be completely impossible.

It is also easy to imagine that a similar problem applies to EACH trigonometric function.

To put it more professionally: these functions are not one-valued, so inverse functions do not exist. In each of the trigonometric functions, each of their values ​​is reached for an infinite number of arguments (they are periodic, right?), so when we try to determine their inverse functions, we will get an infinite number of values ​​​​assigned to each argument. And this cannot be the case in functions.

What to do?

It’s quite simple, not to mention vulgar. Each trigonometric function can be TRUNCATED to obtain a one-valued function.

Let’s get started, let’s define all 4 inverse trigonometric functions correctly:

arcsinx ({sin}^{-1} {x})

Let us recall the graph of the sinx function:

sinx chart

If we agree to cut it, for example, to a compartment , we will get a chart like this:


Unfortunately, this is not what we want, because there is no graph of a one-valued function and a problem with the value, e.g. still occurs:

Graph of the sinx function in the interval [0,pi] with the value 1/2 marked

So we agree that we will trim the sinx function differently, to the arguments : :

Graph of the sinx function for x belonging to [-pi/2,pi/2]

Now it is a one-valued function and there is an inverse arcsinx ({sin}^{-1} {x}) function to it.

The graph of the arcsinx ({sin}^{-1} {x}) function will look something like this:

Graph of the arcsinx function

Its domain is the interval does not exist.

The precise definition of the arcsinx ({sin}^{-1} {x}) function is therefore:


arccosx ({cos}^{-1} {x})

The cosx function is also not a single-valued function:

Graph of the cosx function

However, to obtain a one-valued function, we must trim it to an interval :

Graph of the cosx function truncated to the interval [0,pi]

The function defined in this way is already single-valued and has the inverse function arccosx ({cos}^{-1} {x}).

Its graph will be approximately:

Graph of the arccosx function

And its strict definition:


arctgx ({tan}^{-1} {x})

The tgx function graph looks like this:

Graph of the tgx function

It’s also not a single-valued function! We can cut it as follows:

Graph of the tgx function limited to the range [-pi/2,pi/2]

Thus obtaining a single-valued function.

The arctgx ({tan}^{-1} {x}) function graph looks like this:

Arctgx function graph

And its precise definition is as follows:

, for y\in \left( -\frac{\pi }{2} ,\frac{\pi }{2} \right) .

Let’s also note that the graph shows some interesting properties, e.g.:

  • the domain of the arctgx ({tan}^{-1} {x}) function is the entire set of real numbers (we can calculate arctg from each number)

arcctgx ({cot}^{-1} {x})

From the ctgx function graph:

ctgx function graph

We cut out a multi-valued piece:

Fragment of the ctgx function graph

The arcctgx ({cot}^{-1} {x}) function graph looks like this:

Graph of the arcctgx function

A precise definition of arcctgx ({cot}^{-1} {x}) would be:


It seems:

  • the domain of the arcctgx ({cot}^{-1} {x}) function is the entire set of real numbers (we can calculate arcctg ({cot}^{-1} {x}) from each number)


In many calculators and mathematical notations in general (especially Western ones), inverse trigonometric functions are not marked as “arcus”, but with an exponent of -1. For example, arcsinx is written as . If you know what you’re talking about, there’s no problem. However, you can make a terrible mistake and confuse the inverse of sinx with a function – which is a completely different function from arcsinx.

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