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Intro to arccosine

Sal introduces arccosine, which is the inverse function of cosine, and discusses its principal range. Created by Sal Khan.

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Video transcript

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