when working with inverse trigonometric functions we're also going to be interested in finding inverses of cosine and tangent we'll start with the arc cosine the inverse of cosine same idea as the sine function if we continue the cost function on co starts at its peak then goes down but it would come back up however if we restrict our domain to zero to pi then on that interval the cosine function passes the horizontal line test and so we define this capital c cos function which just limits the domain to that safe interval where we do pass the horizontal line test we can't go any further because then we start getting duplicate y values again can't go any further back for the same reason notice it's a bit awkward in the sense that this interval is not the same interval as we had for the sine function that was negative pi over 2 to pi over 2 it's just the nature of these graphs where is a stretch closest to 0 because we like that where the function is invertible so cosine the interval that we choose is 0 to pi once we have defined that narrow interval for the inver for the cosine function giving it that capital c cos then we can do the same thing we did with the sine function and that is reflected it's a little less obvious what the graph looks like here but you can see that this point at 0 1 gets reflected to the point 1 0 the point pi over 2 0 gets reflected up to 0 pi over 2 and similarly for the final point here at pi negative 1 goes to negative 1 pi and from there you can fill in the graph of the inverse or the arc cosine graph again remembering this inverse this negative 1 here means inverse and not reciprocal it's not 1 over cosine as a quick practice exercise we can use the calculator to find the arc cost of 0.7 again making sure we're in radians because it's a calculus class and we get a value of 0.795 again remembering that this is a ratio of side lengths and this would be an angle and so it would be in radians now what we can do is draw a triangle that would capture that relationship in a more visual way what we can say is we're looking for a ratio of 0.7 and an easy way to do that is we define our adjacent and hypotenuse side so let's do that as our angle theta we're saying the ratio of the adjacent side to the hypotenuse is 0.7 to one if you liked you could write that as seven to ten the beautiful thing about triangles is that it doesn't matter as long as the ratio is consistent all you're doing is scaling the whole size and what we found here as the angle in this particular triangle would be 0.795 radians so the arc cos takes us from ratios of adjacent over hypotenuse to the angle that would be defined in a triangle there now quick question if we do this with arc cosine of 2 let's take a quick stab at that you get a math error when that happens but it seems like a pretty reasonable calculation take a moment to think about why that might be the case why might we get an error when we compute our cos2 with a calculator well one of the things we can rule out pretty quickly is that it's too big a number to handle two is a fine number it might be mathematical issues but that's not the one the inverse characters not understand the business of taking the inverse that doesn't sound right because we're able to do inverses we just did on the last page the cosine does not really have an inverse no that's not true either it does if we limit the domain that was our capital c cosine objective and the issue is really this what we'd be saying here is we have a ratio of adjacent over hypotenuse that's equal to two and so imagine what that would look like as a triangle we would have the adjacent is 2 and the hypotenuse is 1 and that's just not possible with a triangle we cannot build a triangle like that so we can't have an input of two to this particular function the domain of possible ratios for arc cos is going to be negative one to one we have to have the adjacent side smaller than the hypotenuse and any reasonable triangle in the same vein we can find an inverse of the tangent function so the tangent function has this classic asymptotic shape and usually if we had the entire tan function it would repeat itself an infinite number of times as we move to the right however if we want a domain where we pass the horizontal line test we can't include these other branches or the ones that come later because then we'd be crossing the same y value at different x points if we limit ourselves just to this interval here negative pi over 2 to pi over 2 we have one smooth continuous version of the tangent function which does pass the horizontal line test so we define capital t tan of x as the same as tan no difference except we limit ourselves to this negative pi over 2 to pi over 2 interval now that we have a part of the tangent function we can define an inverse sensibly and we do that simply by again reflecting across the y equals x line replacing the the meaning of the x coordinates with the meaning of the y coordinates and what that does is it takes this vertical asymptote and turns it into a horizontal asymptote and likewise on the other side this branch here which approaches x equals negative pi over 2 when we flip it or reflect it now approaches y values of negative pi over two as a quick calculation test what's the value of arctan of one well if we go to one here we get a value and it looks like it's about halfway we can also go to our calculator and when we do that we get 0.785 that's not too indicative the best way to look at this is to again draw a triangle and in this case here this is a ratio and it's the ratio of opposite over adjacent and if that's the case the ratio is one to one so we can draw a triangle that's one by one and then we connect the hypotenuse there well we already know that that's going to be a symmetric triangle here and that means the angles here have to be 45 degrees or exactly halfway up here pi over 4 radians so pi over 4 radians or 0.785 radians same value coming most definitively or most exactly from a sketch of the triangle it's worth making a quick note about the arctan function in particular and some of its unique properties we don't actually have that many functions in our tool kit that when we approach x equals positive infinity or x goes towards positive infinity we have a horizontal asymptote at one value but when we go in the other direction we approach a different horizontal asymptote and it's worth noting that we can make statements about this using our limit terminology if we take the limit as x approaches infinity of this function we're going to get closer and closer to pi over 2. and so in the limit the asymptote value is pi over 2 whereas when we approach it from negative infinity approach towards negative infinity going to the left then this function here this graph is going to get closer and closer and closer to negative pi over 2. so this harks back to what we mean when we talk about limits at infinity and how we can see those values graphically if we have internalized the shapes of the functions that we're working with