 Hi, I'm Zor. Welcome to Unizord Education. This is the first of the mini-series of six lectures about inverse trigonometric functions, and today's subject is arc sign. I do suggest you to go back to Unizord.com and review the introduction to inverse functions. It's just general considerations what they are, et cetera. So let's talk about arc sign. So y is equal to arc sign of x. Now, when we're talking about any function, we have to talk about its domain, its codomain, its range, about whether it's monotonous or not, et cetera. But talking about inverse functions in general and trigonometric functions in particular, we really have to start from the regular trigonometric function. In this case, the regular one is sign, which this one is supposed to be inverse of. Now, is the regular sign an inversible function? Well, again, in the introduction to inverse trigonometric function I was talking about, that none of the trigonometric functions by itself is inversible because it does not establish at one-to-one correspondence between its domain and its range. Now, if this is a sign, for instance, function, then the graph of the sign is like this and domain is all the different real numbers and real number represents an angle in regions and the range is from minus one to minus one. But for any value, let's say one-half of the function, you have many different values of argument where the function takes this value. So there is no one-to-one correspondence between the codomain and the range. And to be able to inverse the function signed in this particular case, we have to reduce the domain to a proper interval where this function is inversible. And the best choice is to find the domain or actually subdomain where the function is monotonous because monotonous functions do establish one-to-one correspondence and completely fill out the range. So what is the interval which can be chosen where the sign function is monotonous and takes all the values within the range? Well, traditionally, it's from minus pi over 2 to pi over 2. So additionally, this piece of the sign function is used to inverse the function and convert it into arc sign. So number one job to inverse the function sign is to reduce it to the interval where it's monotonous and still fills up the range, whatever the range is. In this case, it's minus one-to-one. Once it's chosen, we basically do this. So we have this new function, which we still call sign, but it's the function which is exactly the same as the original sign on minus pi over 2 to pi over 2 interval. And it does not really exist at all. It's not defined outside of this function. I can call it a new sign function or just a sign function, but keep it in mind that whenever we are talking about arc signs, we don't mean exactly this classical sign function, which is the wave for the whole range of real numbers. We only mean the sign function reduced to an interval from minus pi over 2 to pi over 2. Now, why minus pi over 2 to pi over 2? Well, number one, monotonous. Number two, it fills out the range. Well, we can actually choose some other interval, quite frankly. I mean, if this is the continuation of my function, I can choose the interval from pi over 2 to 3 pi over 2. In this interval, the function is also monotonous. It's monotonous and decreasing. And on this interval, it fills out also completely the range. Can it be defined this way? Yes. Traditionally, it's not done this way, just because it seems to me a little bit analog. Easier, simpler, it's around zero. I mean, obviously, any piece of the real numbers axis, the x-axis, where the function is monotonous and fills out the range can be used as a domain of the new sign function, which is inversible in that particular range. But it's not done just traditionally. There's no other consideration except that it looks better. It's closer to zero. And it's still odd, by the way, which kind of easier to deal with. Fine. As soon as we have defined our new function sign, which is defined only in this interval from minus pi over 2 to pi over 2, now we can inverse it and now we can define the function arc sign. So its range is whatever the domain for the sign function is. Its domain is whatever the range of the sign was. So basically, x belongs to an interval for this function from minus 1 to 1. And y, therefore, belongs to an interval from minus pi over 2 to pi over 2. Graph is extremely important. It's really a great tool to see whether the function is monotonous, what's the range, what's the domain, et cetera. So now we'll use the same graph to basically draw the function arc sign x, the same technique. Now, the arc sign x is defined on this interval of minus 1 to 1. Now its range is minus pi over 2 to pi over 2. Now, function sign at 0 is equal to 0, which means sine of 0 is 0. So arc sine of 0 is 0. So our function must actually go through this particular point. Now, in this case, whenever my sine is equal to minus pi over 2, arc sine is equal to minus 1. Sorry, sine is equal to minus 1. So arc sine would be just the other way around. From the point minus 1, it has the value of minus pi over 2. What is the angle sine of which is equal to minus 1? Because that's what arc sine actually means. If you have defined something which is the value of the sine function, then this function returns the value of its argument. If you have defined that minus 1 is the value of the sine function, then minus pi over 2 is the angle its argument. So that's why arc sine actually points here. Similarly, here, so what's the angle if its sine is equal to 1? Well, pi over 2. Look at this graph. We are talking only about this interval, right? I'm not talking about all angles sine of them is equal to 1. I'm talking only the angle in this particular interval where the new sine function is defined. And here, the function actually goes very similar to this one, except if you remember the inverse function is, graph of the inverse function is symmetrical to the graph of the main function relative to the main angle bisector. So the function actually would be curved in this way. This function curves this way, and this function curves this way. So that's the graph. We are talking about the main. We're talking about the range of the function. That's the graph. That's the arc sine, arc sine of x. And one little detail, you see sine function is odd as we know. If you change the sine of the argument, the function changes the sine, right? If you have an angle, let's say pi over 4, then the sine is equal to square root of 2 over 2. If it's minus pi over 4, the sine is equal to minus square root of 2 over 2. So now, here is my question. If the main function is odd, does it mean that inverse function is odd as well? Well, the answer is yes. And here is the proof. It's interesting that we can actually prove something like this. And obviously, I prefer it. So how can we prove that this function is odd? Well, basically, I have to prove that arc sine of minus x is equal to minus arc sine of x. That's the definition of the odd function. You change the argument, then the function changes the sine as well. Now, how can I prove this? How can I prove that arc sine of minus x equals to a? How can I prove it? What does it mean that arc sine of minus x is equal to a? What arc sine means? It means what's the angle sine of which is equal to whatever the argument is, right? So that's the definition of arc sine. So I have to find an angle a sine of which is equal to minus x. So this is equivalent to sine of this should be equal to this, should be equal to minus x. So let's check it out, minus. So if I will show that the sine of this is equal to minus x, it actually means, by definition of this arc sine, that this equality holds. Now, let me just make it more visible, minus arc sine of x. Now, what is sine of minus arc sine of x? Well, we know that the sine is an odd function, right? So if I change the value of the argument to an opposite, then the value of the function itself is changing, right? So it's minus sine of arc sine of x, right? This is because sine is an odd function. If I change the sine of whatever argument of the sine is, then the function itself changes the sine. Now, what is this? Well, by definition of the arc sine, this is x. Because what is arc sine of x? Arc sine of x is an angle sine of which is equal to x. Well, let's change the sine of this angle. Then we will get x. So we got this. So what it means is if the sine of something is minus x, it means, by definition of the arc sine, that arc sine of this is equal to whatever the angle here is, which is this equality. So that's the proof. That's the end of the proof action. So function arc sine is odd. Well, that's basically it. There are certain problems about arc signs and as well as other trigonometric functions, which I will present in due time. But this is a small, very short lecture about just properties of the function arc sine. And I do recommend you to, again, go through unizor.com notes for this lecture. Read it. And well, if you understand everything, that's great. And then go to the next one. It's six lectures in this series for each arc function, each inverse function, which exists. So thank you very much and good luck.