Using inverse trig functions with a calculator

Javier is calibrating sophisticated medical imaging equipment. The manual reports that the tangent of a particular angle is one. So that's saying that the tangent and let's say that that particular angle is theta is equal to one. What should Javier do to find the angle? And I encourage you to pause this video and look at these choices and think about which of these should he do to find the angle? So let's look through each of them. So the first one... well, actually, instead of looking at the choices, let's think about what we would do to find the angle. So they're saying that the tangent of some angle is equal to one. Well one thing that you might want to do is say okay, if we take the inverse tangent, if we take the inverse tangent, of the tangent of theta, so if we take the inverse tangent of both sides of this, we of course would get the inverse tangent of the tangent of theta. If the domain over here is restricted appropriately, is just going to be equal to theta, so we could say the theta is going to be the inverse tangent of one. So it might be tempting to just pick this one right over here. Type inverse tangent of one into his calculator. So maybe this looks like the best choice. But remember, I said if we restrict the domain right over here. if we restrict the possible values of tangent, of theta here appropriately, then this is going to simplify to this. But there is a scenario where this does not happen. And that's if we pick thetas that are outside of the range of the inverse tangent function. What do I mean by that? Well it's really just based on the idea that there are multiple angles that have... or multiple angles whose tangent is one. And let me draw that here with a unit circle here. So we draw a unit circle, so that's my x axis, that's my y axis, let me draw my unit circle here. Actually you probably don't even have to draw the unit circle, because the tangent is really much more about the slope of the ray created by the angle, than where it intersects the unit circle as would be the case with sine and cosine. So if you have.. so you could have this angle right over here. So let's say this is a candidate theta, where the tangent of this theta is the slope of this line, and this terminal angle, the terminal ray, you could say of the angle. The other side, the initial ray, is along the positive x axis. And so you could say, okay the tangent of this theta, the tangent of this theta is one. Because the slope of this line is one. Let me scroll over a little bit. Well, so let me write it this way. So tangent theta is equal to one. But I can construct another theta whose tangent is equal to one by going all the way over here and essentially going in the opposite direction but the slope of this line, so let's call this theta two, tangent of theta two is also going to be equal to one. And of course you could go another pi radiance and go back to the original angle, but that's functionally the same angle in terms of where it is relative to the positive x axis, or what direction it points into, but this one is fundamentally a different angle. So we do not know, we do not have enough information just given what we've been told to know exactly which theta we're talking about, whether we're talking about this orange theta or this mauve theta. So I would say the get more information, there are multiple angles which fit this description.