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# Intro to arcsine

## Video transcript

if I were to walk up to you on the street and say you please tell me what it's not I didn't 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 would 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 the y coordinate on the unit circle so you would just want to know this value right here and you would immediately see okay 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 degrees this is 90 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 a 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 1/2 x squared is equal to 1 x squared is equal to 1/2 X is equal to the square root of 1/2 which is 1 over the square root of 2 I can put that in rational form by multiplying that by the square root of 2 over 2 time this 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 2 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 in the in the in the 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 please tell me what the arc sine arcsine of the square root of two over two 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 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 two over two all this is asking is what angle what I have to take the sine of in order to get the value square root of two over two this is also asking what angle what I have to take the sine of in order to get square root of two over two I could rewrite either of these statements I can rewrite either of these statements as saying square let me do it I could rewrite either of these statements as saying sign of what is equal to the square root of two over two and this I think is a much easier question for you to answer sign 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 you know 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 arc sine sorry arc sine of square root of 2 over 2 is equal to PI over 4 now you might say so just as a review I'm giving you a value and I'm saying give me an angle that gives me when I taken the sine of that angle that gives me that value but you 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 on 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 sign of any of those angles you can just keep adding 360 degrees if you take the sign 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 ax or 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 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 will dis restrict its range to the most natural place so let's restrict its range and actually I'll just as a as a side note what's its domain restricted to so if I'm taking the arc sine of something so if I'm taking the arc sine 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 have to restrict the range and for arc sine 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 greater than or equal to minus PI over 2 so given that we now understand what arc sine is let's do let's do another problem let me clear out some space here let me do another arc sine so let's say I were to ask you what the arc sine the arc sine of minus square root of 3 over 2 is minus square root of 3 over 2 now you might have that memorized 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 or so let me just draw my unit circle and what I'm dealing with arc sine 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 it means we are right about there so this is minus square root of 3 over 2 this is where we are now what angle gives me that 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 we're going below the x-axis in the clockwise direction and to figure out let me just draw a little triangle here like a better color than that so 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 data 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 minus because we're going down but let's just figure out this angle and what we know it's we know it's a negative angle so when you see square root of 3 over 2 hopefully you recognize hey this is a 30-60-90 triangle the square root of 3 over to the side is 1/2 and then of course this side is 1 because this is a unit circle so it's radius is 1 so in a 30-60-90 triangle the side opposite to the square root of 3 over 2 is 60 degrees the 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 downward 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 time 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 arc sine the arc sine of minus square root of 3 over 2 is equal to minus PI over 3 radians or we could say the inverse sine 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 for second mode I'm in Radian mode so I know I'm gonna get hopefully the right answer and I want to figure out the inverse sine so the inverse sine is second and the sine button of the - square root of 3 over 2 it equals C - 1.04 so it's telling me that this is equal to 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