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# Intro to arctangent

## Video transcript

in the last video I showed you that if someone were to walk up to you and you asked you what is the arcsine whoops arc sine of X and you know so this is going to be equal to who knows what this is just the same thing as saying that look the sine of some angle is equal to X and we solved it a couple of cases in the last example so using the same pattern let me let me say this I could have we also rewritten this as the inverse sine of X is equal to what these are equivalent statements two ways of writing the inverse sine function this is more this is the inverse sine function you're not taking this to the negative one power you're just saying the sine of what of so what question mark what angle is equal to X and we did this in the last video so by the same pattern if I were to walk up to you on the street and I were to say you know the tangent of the the inverse tangent of X is equal to what you should immediately in your head say oh he's just asking me he's just saying look the tangent of some angle is equal to X and I just need to figure out what that angle is so let's do an example so let's say I were to walk up to you on the street there's a lot of walking up on a lot of street trains and see I would have right and I say you what is the arc tangent of minus one or I could have equivalently asked you what is the inverse tangent of minus one these are equivalent questions and what you should do is you should in your head if you don't have this memorize you should draw the unit circle and actually let me just do a refresher of what tangent is even asking us the tangent of theta this is just a straight up vanilla non inverse function tangent that's equal to the sine of theta over the cosine of theta and the sine of theta is is the Y value on the unit function on the unit circle and the cosine of theta is the x value and so if you draw a line let me draw a little unit circle here so if I have a unit circle like that and let's say I'm at some angle let's say that's my angle theta and this is my Y my coordinate it's X Y we know already that the Y value this is the sine of theta let me scroll over here sine of theta and we already know that this x value is the cosine of theta so what's the tangent going to be it's going to be this distance it's going to be this distance divided by this distance or from your algebra one this might strike a bell because we're starting at the origin from the point zero zero this is our change in Y over our change in X or it's our rise over run or you can kind of view the tangent of theta or it really is as the slope of this line the slope so you could write slope is equal to the tangent of theta so let's just bear that in mind when we go to our example if I'm asking you if I'm asking you and I'll rewrite it here what is the inverse tangent of minus one and I'll keep rewriting it or the arctangent of minus one I'm saying what angle gives me a slope of minus one on the unit circle so let's draw the unit circle let's draw the unit circle like that and then I have my axes like that and I want to slope of minus one a slope of minus one looks like this it looks like that alright if it was like that would be slope of +1 so what angle is this so in order to have a slope of minus one this distance is the same as this distance and you might already recognize as you know this is a right angle so these angles have to be the same and so this has to be a 45-45-90 triangle this is an isosceles triangle these two have to add up to 90 and have to be the same so this is 45 45 90 and if you know your 45459 actually you don't even have to know the sides of it a we in the previous video we saw that you know this is going to be right here this this distance is going to be square root of 2 over 2 or so this coordinate and the Y Direction is minus square root of 2 over 2 and there's this coordinate right here on the X Direction is square root of 2 over 2 because this length right there is that so square root of 2 over 2 squared plus the square root of 2 over 2 squared is equal to 1 squared but the important thing to realize this is a 45-45-90 triangle so this angle right here is well if you're just looking at the triangle by itself you would say that this is a 45 degree angle but since we're going clockwise below the x-axis we'll call this a minus 45 degree angle minus 45 degree so that the tangent of minus 40 let me write that down so if I'm in degrees and that tends to be how I think so I could write the tangent the tangent of minus 45 degrees it equals this negative value minus square root of 2 over 2 over square root of 2 over 2 which is equal to minus 1 or I could write the arctangent of minus 1 is equal to minus 45 degrees now if we're dealing with radians we just have to convert this to radians so we multiply that times we get PI radians for every 180 degrees the degrees cancel out you have a 45 over 180 this goes 4 times so this is equal to you have the minus sign minus PI over 4 radians so the arctangent of minus 1 is equal to minus PI over 4 or the inverse tangent the inverse tangent of minus 1 is also equal to minus PI over 4 now you could say look if I'm at minus PI over 4 that's there that's fine this is this gives me a a value of minus 1 because the slope of this line is minus 1 but I could keep going around the unit circle I could add 2 pi to this maybe I could add you know I could add 2 pi to this and that would also give me if I take the tangent of that angle it would also give me minus 1 or I could add 2 pi again again give me minus one in fact I could go to this point right here and the tangent would also give me minus one because the slope is right there and like I said in the sign in the inverse sine video you can't have a function that has a one-to-many map and you can't you know tangent tangent inverse of X can't map to a bunch of different values you can't map to minus PI over four minus PI over four can't map to you know three what it would be 3 PI over 4 I don't know it would be well I'll just say 2 pi minus PI over 4 or 4 PI minus PI I can't map to all of these different things so I have to constrict the range on the inverse tan function and it will restrict it very similarly to the way we restricted the the sine inverse sine range we're going to strict it to the first and fourth quadrants so the answer to your inverse tangent is always going to be something in these quadrants but it can't be this point in that point because the tangent function becomes undefined at PI over at PI over 2 and at minus PI over 2 because your slope goes vertical you start dividing your change in X is 0 you're dividing your cosine of theta goes to 0 so if you divide by that it's it's undefined so your range so if I let me write this down so if I have an inverse tangent of X I'm going to well what are all the values that the tangent can take on so if I have the tangent of theta is equal to X what are all the different values that X could take on these are all the possible values for the slope and that slope can take on anything so X could be anywhere between minus infinity and positive infinity X can pretty much take on any value but what about theta well I just said it theta you can only go from minus PI over 2 all the way to PI over 2 and you can't even include PI over 2 or minus PI over 2 because then you'd be vertical so then you say or so if I'm just dealing with vanilla tangent not the inverse the domain well that well the domain of tangent can go multiple times around so let me not make that statement but if I want to do inverse tangent so I don't have a one-to-many mapping I want to cross out all of these I'm going to restrict theta or my range to be greater than minus PI over two and less than positive PI over two and so if I restrict my range to this right here and I exclude that point and that point then I can only get one answer when I say what tangent of what gives me a slope of minus one and that's the question I'm asking right there there's only one answer because if I keep this one falls out of it and obviously as I go around and around those fall out of that valid range for theta that I was giving you and then let you know just to kind of make sure we did it right we our answer was PI over four let's see if we get that when we use our calculator so the inverse tangent the inverse tangent of -1 is equal to that let's see if that's the same thing as minus PI over 4 minus PI over 4 is equal to that so it is minus PI over 4 but it was good that we solved it without the calculator because it's hard to recognize this as minus PI over 4