The inverse trigonometric functions
Intro to arcsine
If I were to walk up to you on the street and say you, please tell me what-- so I didn't want to write that thick --please tell me what sine of pi over 4 is. And, obviously, we're assuming we're dealing in radians. You either have that memorized or you would draw the unit circle right there. That's not the best looking unit circle, but you get the idea. You'd go to pi over 4 radians, which is the same thing as 45 degrees. You would draw that unit radius out. And the sine is defined as a y-coordinate on the unit circle. So you would just want to know this value right here. And you would immediately say OK. This is a 45 degrees. Let me draw the triangle a little bit larger. The triangle looks like this. This is 45. That's 45. This is 90. And you can solve a 45 45 90 triangle. The hypotenuse is 1. This is x. This is x. They're going to be the same values. This is an isosceles triangle, right? Their base angles are the same. So you say, look. x squared plus x squared is equal to 1 squared, which is just 1. 2x squared is equal to 1. x squared is equal to 1/2. x is equal to the square root of 1/2, which is one over the square root of 2. I can put that in rational form by multiplying that by the square root of 2 over 2. And I get x is equal to the square root of 2 over 2. So the height here is square root of 2 over 2. And if you wanted to know this distance too, it would also be the same thing. But we just cared about the height. Because the sine value, the sine of this, is just this height right here. The y-coordinate. And we got that as the square root of 2 over 2. This is all review. We learned this in the unit circle video. But what if someone else-- Let's say on another day, I come up to you and I say you, please tell me what the arcsine of the square root of 2 over 2 is. What is the arcsine? And you're stumped. You're like I know what the sine of an angle is, but this is some new trigonometric function that Sal has devised. And all you have to realize, when they have this word arc in front of it-- This is also sometimes referred to as the inverse sine. This could have just as easily been written as: what is the inverse sine of the square root of 2 over 2? All this is asking is what angle would I have to take the sine of in order to get the value square root of 2 over 2. This is also asking what angle would I have to take the sine of in order to get square root of 2 over 2. I could rewrite either of these statements as saying square-- Let me do it. I could rewrite either of these statements as saying sine of what is equal to the square root of 2 over 2. And this, I think, is a much easier question for you to answer. Sine of what is square root of 2 over 2? Well I just figured out that the sine of pi over 4 is square root of 2 over 2. So, in this case, I know that the sine of pi over 4 is equal to square root of 2 over 2. So my question mark is equal to pi over 4. Or, I could have rewritten this as, the arcsine-- sorry --arcsine of the square root of 2 over 2 is equal to pi over 4. Now you might say so, just as review, I'm giving you a value and I'm saying give me an angle that gives me, when I take the sine of that angle that gives me that value. But you're like hey Sal. Look. Let me go over here. You're like, look pi over 2 worked. 45 degrees worked. But I could just keep adding 360 degrees or I could keep just adding 2 pi. And all of those would work because those would all get me to that same point of the unit circle, right? And you'd be correct. And so all of those values, you would think, would be valid answers for this, right? Because if you take the sine of any of those angles-- You could just keep adding 360 degrees. If you take the sine of any of them, you would get square root of 2 over 2. And that's a problem. You can't have a function where if I take the function-- I can't have a function, f of x, where it maps to multiple values, right? Where it maps to pi over 4, or it maps to pi over 4 plus 2 pi or pi over 4 plus 4 pi. So in order for this to be a valid function-- In order for the inverse sine function to be valid, I have to restrict its range. And the way that-- We'll just restrict its range to the most natural place. So let's restrict its range. Actually, just as a side note, what's its domain restricted to? So if I'm taking the arcsine of something. So if I'm taking the arcsine of x, and I'm saying that that is equal to theta, what's the domain restricted to? What are the valid values of x? x could be equal to what? Well if I take the sine of any angle, I can only get values between 1 and negative 1, right? So x is going to be greater than or equal to negative 1 and then less than or equal to 1. That's the domain. Now, in order to make this a valid function, I have to restrict the range. The possible values. I have to restrict the range. Now for arcsine, the convention is to restrict it to the first and fourth quadrants. To restrict the possible angles to this area right here along the unit circle. So theta is restricted to being less than or equal to pi over 2 and then greater than or equal to minus pi over 2. So given that, we now understand what arcsine is. Let's do another problem. Clear out some space here. Let me do another arcsine. So let's say I were to ask you what the arcsine of minus the square root of 3 over 2 is. Now you might have that memorized. And say, I immediately know that sine of x, or sine of theta is square root of 3 over 2. And you'd be done. But I don't have that memorized. So let me just draw my unit circle. And when I'm dealing with arcsine, I just have to draw the first and fourth quadrants of my unit circle. That's the y-axis. That's my x-axis. x and y. And where am I? If the sine of something is minus square root of 3 over 2, that means the y-coordinate on the unit circle is minus square root of 3 over 2. So it means we're right about there. So this is minus the square root of 3 over 2. This is where we are. Now what angle gives me that? Let's think about it a little bit. My y-coordinate is minus square root of 3 over 2. This is the angle. It's going to be a negative angle because we're going below the x-axis in the clockwise direction. And to figure out-- Let me just draw a little triangle here. Let me pick a better color than that. That's a triangle. Let me do it in this blue color. So let me zoom up that triangle. Like that. This is theta. That's theta. And what's this length right here? Well that's the same as the y-height, I guess we could call it. Which is square root of 3 over 2. It's a minus because we're going down. But let's just figure out this angle. And we know it's a negative angle. So when you see a square root of 3 over 2, hopefully you recognize this is a 30 60 90 triangle. The square root of 3 over 2. This side is 1/2. And then, of course, this side is 1. Because this is a unit circle. So its radius is 1. So in a 30 60 90 triangle, the side opposite to the square root of 3 over 2 is 60 degrees. This side over here is 30 degrees. So we know that our theta is-- This is 60 degrees. That's its magnitude. But it's going downwards. So it's minus 60 degrees. So theta is equal to minus 60 degrees. But if we're dealing in radians, that's not good enough. So we can multiply that times 100-- sorry --pi radians for every 180 degrees. Degrees cancel out. And we're left with theta is equal to minus pi over 3 radians. And so we can say-- We can now make the statements that the arcsine of minus square root of 3 over 2 is equal to minus pi over 3 radians. Or we could say the inverse sign of minus square root of 3 over 2 is equal to minus pi over 3 radians. And to confirm this, let's just-- Let me get a little calculator out. I put this in radian mode already. You can just check that. Per second mode. I'm in radian mode. So I know I'm going to get, hopefully, the right answer. And I want to figure out the inverse sign. So the inverse sine-- the second and the sine button --of the minus square root of 3 over 2. It equals minus 1.04. So it's telling me that this is equal to minus 1.04 radians. So pi over 3 must be equal to 1.04. Let's see if I can confirm that. So if I were to write minus pi divided by 3, what do I get? I get the exact same value. So my calculator gave me the exact same value, but it might have not been that helpful because my calculator doesn't tell me that this is minus pi over 3.