 If you have ever tried to use the inverse functions of cosine and sine on your calculator, you might have come across the following problem. Your calculator gave you an answer, but then that answer was not the correct answer. So, what is going on here? Is your calculator lying to you? No, the calculator is not lying to you. What happens with the inverse cosine and sine function is that the calculator only gives you one solution. However, there are multiple solutions and that one solution, maybe depending on the problem, is not the one solution you're looking for in the problem. Let me give you an example. So, let's say you have the following problem. You are given a function of the form x as a function of time is three times cosine of 0.1 t plus pi. In physics, this could be the equation of a simple harmonic motion. And the question is the following at what time bigger than zero is x equals 2.5. So, you might have started solving this thinking, okay, this is easy. So, all I have to do is I have to plug in right 2.5 is equal to 3 times cosine 0.1 t plus pi. Then you went on and you solved for your time. So, we did 2.5 over 3 is cosine of 0.1 t plus pi. And then you know if you want to do the inverse of cosine, you have to do the r cosine on the other side. So, we have r cosine 2.5 over 3 is 0.1 t plus pi. Now, after hours of looking for the r-course button on your calculator, you have finally figured out that this one is actually labeled cosine minus one. And then you type this in and what the calculator will tell you that the inverse cosine of 2.5 over 3 is the calculator will tell you that this is 0.585, which you said equal to 0.1 times time plus pi. So, you solve this for time, you subtract pi, you divide by 0.1. And what you get is time is equal to minus 26 ish seconds. Now, you submit that to your teacher and your teacher goes this is wrong because I want the t is bigger than 0. So, what is going on here and how can we solve this? Now, what you have to realize first is by asking the calculator to give you the cosine inverse function of 2.5 over 3, you're actually asking the calculator for which angle, if I do cosine of that angle, we get 2.5 over 3. 2.5 over 3, if you don't really like this, that's by the way, that is equal to 0.83. So, the cosine function looks as following. If we plot cosine of an angle theta as a function of theta, we will start at 1, go down to 0, come back to 1, and so on. And it will also extend on the left side. Now, what you're basically asking your calculator by typing this one here is, okay, so for what angle do I get 0.83? So, 0.83 is about here. Now, before we had calculators, what people had, they actually have printouts of the cosine function, they will put the ruler down here, and then they would immediately see, oh, look, I have multiple solutions. I have one here, I have one here, I have one here. Now, your calculator, however, it keeps things simple. It will not give you all the solution, all the calculator does, it gives you the solution between here and here. The calculator just gave you was that we're going to get one solution, which is theta 1, which is, in this case, 0.585 rats, because I had set my calculator to radians, so he spit me out the answer in radians. If you put it in degrees, the calculator, of course, will give you the answer in degrees, but the problem here is exactly the same, you will only get one angle, but there are several possible solutions. So now, how do we get the other solutions? Well, the key to that here is a sketch, and you look at the symmetries of the sketch. Here, I had another one, let's call this theta 0, that will simply be my other angle inverse. So, this one here was theta 1, which led me to a time 1, then I had theta 0, which is the same number, but negative, minus 0.585 equals to 0.1t plus pi, and then I will get a t0, which is equal to, what will I get, minus 37 seconds. That's even worse than my first one, I even get more negative. So, what about that one here? This call is theta 2. How do I get theta 2? If you look at the distance, well, there are several ways of solving it. One of them is, for example, if you look here, the link between my theta 0 and theta 2 is plus 2 pi rat, if you're calculating invadence, or 360 degrees, if you're calculating in degrees, or if ever you're actually calculating in the time domain, this will be plus 1 period. So, the period here has an angle either in 2 pi rats or 360 degrees. So, what I could do is I can do, okay, my theta, this was theta naught, so this my theta 2, would be my minus 0.585 plus 2 pi equals 0.1t plus pi, and then I solve this and I will get my third time is plus 26 seconds, which will be my first positive answer, so the answer to this problem. So, before I mention something about the period, if this is simple harmonic motion, then we can calculate the period as being 2 pi over my angular frequency, which is the 0.1, so 2 times 3.14 divided by 0.1, that gives me about 63 seconds. Now, I could check, according to my sketch here, my result that I got from my second angle would be my 0th angle, the one before, plus 2 pi, or for times, the one period would be the 63 seconds. So, is this plus 63 seconds? And it is. So, what was to be shown, quote, era demonstrandum. So, little rewrap, your calculator for cosine will only give you the one answer between 0 and pi rat, or 180 degrees, if you're in degrees. Similarly, the sine function, for the sine function, your calculator will always just spit you out the answer between minus pi half or 90 degrees and plus pi half at 90 degrees. And similar as we just did with the cosine, what you always have to do is make yourself a little sketch of this and figure out where are my other solutions. For example, here I had one, here I had one, and then use the symmetries in your cosine or sine functions, use that the function repeats itself at fixed intervals to figure out what are the other solutions, and then you have to think which one is the solution that actually solves your problem.