Intro to arccosine

I've already made videos on the arc? sine and the arc tangent, so to kind?? of complete the trifecta I might as well? make a video on the arc cosine and just?? like the other inverse trigonometric? functions the arc cosine it's kind of?? the same thought process if I were to? tell you that the arc now I'm doing?? cosine if I were to tell you that the? arc cosine of X is equal to theta this?? is an equivalent statement to saying? that the inverse cosine of X is equal?? to theta these are just two different? ways of writing the exact same thing and?? as soon as I see either an arc anything? or an inverse trig function in general?? my brain immediately rearranges this my? brain regionally immediately says this?? is saying that if I take the cosine of? some angle theta that I'm going to get X?? or if I get in order the same statement? up here either of these should boil down?? to this if I say that the coast you know? what is the inverse cosine of X my brain?? says what angle can I take the cosine of? to get X so with that said let's try it?? out on an example let's say that I have? the arc I'm told no I put two CS there?? I'm told to evaluate the arc cosine of? minus 1/2 my brain as you know let's say?? that this is going to be equal to its? going to be equal to some angle and this?? is equivalent to saying that the cosine? of my mystery angle is equal to minus?? 1/2 and as soon as you put it in this? way at least for my brain it becomes a?? lot easier to process so let's draw our? unit circle and see if we can make some?? headway here so that's my let me see I? could draw a little straighter actually?? maybe I could actually draw put rulers? here and if I put a ruler here maybe?? I can draw a straight line let me see? no that's too hard okay so that is my?? y-axis that is my x-axis not the neatest? most neatly drawn axes but it'll do and?? let me draw my unit circle looks more? like a unit ellipse but you get the?? idea and the cosine of an angle is a? defined on the unit circle definition?? is the x-value on the unit circle so if? we have some some angle the x-value is?? going to be equal to minus 1/2 so we? go to minus 1/2 right here and so the?? angle that we have to solve for R theta? is the angle that when we intersect the?? unit circle the x value is minus 1/2 so? let me see this is the angle that we're?? trying to figure out this is theta that? we needed to determine so how can we do?? that so if this is minus 1/2 right here? let's figure out these different angles?? and the way I like to think about it is? I like to figure out this angle right?? here and if I know that angle I can? just subtract that from 180 degrees?? to get this this light blue angle that's? kind of the solution to our problem so?? let me make this triangle a little bit? bigger so that triangle now let me do it?? like this that triangle looks something? like this where this distance right here?? is 1/2 that distance right there is 1/2? this distance right here is 1 hopefully?? you recognize that this is going to be? a 30-60-90 triangle you could actually?? solve for this other side you'll get to? square root of 3 over 2 and to solve for?? that other side you just need to do the? Pythagorean theorem actually let me just?? do that let me just call this I don't? know let me just call this a so you'd?? get a squared plus 1/2 squared which is? 1/4 is equal to 1 squared which is 1 you?? get a squared is equal to 3/4 or a is? equal to the square root of 3 over 2 so?? you immediately notice 30-60-90 triangle? and you know that because the sides of?? a 30-60-90 triangle if the hypotenuse is? 1 or 1/2 and square root of 3 over 2 and?? you'll also know that the side opposite? the square root of 3 over 2 side is the?? 60 degrees that's 60 this is 90 this is? the right angle and this is 30 right up?? there but this is the one we care about? this angle right here we just figured?? out is 60 degrees so what's this what's? the bigger angle that we care about what?? is 60 degrees supplementary to it's? supplementary to 180 degrees so the?? arc cosine or the inverse cosine let me? write that down the arc cosine of minus?? 1/2 is equal to 100 and 120 degrees I'll? write 180 there no it's 180 minus the 60?? this whole thing is 180 so this is right? here is 120 degrees right 120 plus 60?? is 180 or if we wanted to write that in? radians you just write 120 degrees times?? pi Radian per 180 degrees degrees cancel? out 12 over 18 is 2/3 so it equals 2 PI?? over 3 radians so this right here is? equal to 2 pi PI over 3 radians now?? just like we saw in the arc sine and? the arc tangent videos you probably say?? hey okay if I have 2 PI over 3 radians? that gives me a cosine of minus 1/2 and?? I could write that cosine of 2 pi over? 3 is equal to minus 1/2 this gives you?? the same information as the statement? up here but I could just keep going?? around the unit circle for example I? could I will at this point over here?? cosine of this angle if I were to add? if I were to go this far would also?? be minus 1/2 and then I could go 2 pi? around and get back here so there's a?? lot of values that if I take the cosine? of those angles I'll get this minus 1/2?? so we have to restrict ourselves we? have to restrict the values that the?? arc cosine function can take on so we're? essentially restricting its range we're?? restricting its range what we do is? we restrict the range to this upper?? a hemisphere the first and second? quadrants so if we say if we make?? the statement that the arc cosine? of X is equal to theta we're going?? to restrict our range theta to that top? so theta is going to be greater than or?? equal to zero and less than or equal? to 102 PI less oh sorry not 2pi less?? than or equal to PI right or this is? also zero degrees or 180 degrees we're?? restricting ourselves to this part of? the hemisphere right there and so you?? can't do this this is the only point? where the cosine of the angle is equal?? to minus 1/2 we can't take this angle? because it's outside of our range and?? what are the valid values for X well? any angle if I take the cosine of it?? it can be between minus 1 and plus? 1 so X the domain for the the domain?? for the our cosine function is going? to be X has to be less than or equal?? to 1 and greater than or equal to minus? 1 and once again let's just check our?? work let's see if if the value I got? here that the arc cosine of minus 1/2?? really is 2 PI over 3 as calculated by? the ti-85 let me turn it on so I need?? to figure out the inverse cosine which? is the same thing as the arc cosine of?? minus 1/2 of minus 0.5 it gives me that? decimal that strange number let's see?? if that's the same thing is 2 PI over? 3 2 times pi divided by 3 is equal to?? that exact same number so the calculator? gave me the same value I got but this is?? kind of a useless what's not a useless? number it's it's a valid that's that is?? the answer but it's it doesn't it's not? a nice clean answer I didn't know that?? this is 2 PI over 3 radians and so when? we did it using the unit circle we were?? able to get that answer so hopefully? and actually let me ask you let me?? just finish this up with an interesting? question and this applies let's do all?? of them if I were to ask you you know? say I were to take the arc arc cosine?? of X and then I were to take the cosine? of that what is this what is this going?? to be equal to well this statement right? here could be said well let's say that?? the arc cosine of X is equal to theta? that means that the cosine of theta is?? equal to X right so if the arc cosine? of X is equal to theta we can replace?? this with theta and then the cosine? of theta well the cosine of theta is?? X so this whole thing is going to be? X hopefully I didn't confuse you there?? right I'm just saying look R cosine? of X let's just call that theta now?? it by definition this means that the? cosine of theta is equal to X these?? are equivalent statements these are? completely equivalent statements right?? here so if we put a theta right there? we take the cosine of theta has to be?? equal to X now let me ask you a bonus? slightly trickier question what if I?? were to ask you and this is true for any? X that you put in here this is true for?? any X any value between negative 1 and? 1 including those two endpoints this is?? going to be true now what if I were to? ask you what the arc arc cosine of the?? cosine of theta is what is this going? to be equal to my answer is it depends?? it depends on the theta so if theta is? in the if theta is in the range if theta?? is between if theta is between 0 and pi? so it's in our valid range for for kind?? of our range for the product of the arc? cosine then this will be equal to theta?? if this is true for theta but what if we? take some data out of that range let's?? try it out let's sake so let me do it? 1 with theta in that range let's take?? the arc cosine of the cosine of let's? just do some one of them that we know?? let's take the cosine of let's take? the cosine of 2 pi over 3 cosine of 2?? pi over 3 radians that's the same thing? as the arc cosine of minus 1/2 cosine?? of 2 pi over 3 is minus 1/2 we just saw? that in the earlier part of this video?? and then we solve this we said oh this? is equal to 2 PI over 3 so if we're in?? the range if theta is between 0 and? pi it worked and that's because the?? arc cosine function can only produce? values between 0 and PI but what if?? I were to ask you what is the arc arc? cosine of the cosine of I don't know?? of 3 PI of 3 PI so if I were to draw? the unit circle here let me draw the?? unit circle real quick one and that's my? axes what's 3 pi 2 pi is if I go around?? once and then I go around another pi so? I end up right here so I've gone around?? one and a half times the unit circle so? this is at 3 pi what's the x-coordinate?? here it's minus one so cosine of 3 pi is? minus one all right so what's what's arc?? cosine of minus one arc cosine of minus? one well remember the the range or the?? set of values that are cosine can be can? evaluate to is in this upper hemisphere?? it's between this can only be between PI? and 0 so arc cosine of negative one is?? just going to be PI so this is going? to be PI our cosine of negative this?? is this is negative one our cosine? of negative one is PI and that's?? a reasonable statement because the? difference between 3 PI and PI is just?? going around the unit circle a couple? of times and so you get an equivalent?? it's kind of your the equivalent point? on the unit circle so I just thought I?? would throw those two at you this one I? mean this is a useful one if I actually?? let me write it up here this one is a? useful one the cosine of the arc cosine?? of X is always going to be X I can so do? that with sign the sign of the arc sine?? of X is also going to be X and these are? just useful things to you shouldn't just?? memorize them because obviously you? might memorize it the wrong way but?? you just think a little bit about? it and it you'll never forget it