For this homework assignment you will definitely need a piece of paper and pen or pencil and, first of all, please, please, please watch these lectures. Well, they're not even lectures, 1 minute and 42 second of a brief overview of the idea and then 1 minute, 29 seconds sample and that will help you to save a lot of time. All problems of this type are a little bit the same, so you can get them all done in like what, 6 steps or something. How many steps do I got? [Inaudible] over here, let me see, 1, 2, 3, 4, 5, there is 6; yep, 6 steps, right? So, let's-- so let me show you a few examples. Let's just jump right into the homework and the homework looks like this. This is one of the most crucial skills necessary for calculus II from trigonometry, and students frequently by the time they get to the part of the calculus II that requires this, they frequently forget this. So, please, please, please review this carefully. We're looking for the sin of cotangent inverse. Cotangent inverse is also known as arccotangent, and I actually prefer that name, but the idea is very simple; we can call what's inside of the parentheses here, theta. The problem becomes quite simple. It's just finding sin of theta. Sin of theta it's also very simple. It's just the opposite over adjacent. Now,, in order for us to understand what the opposite one is adjacent, we have to have a triangle. So, in this triangle this is theta, this is going to be my opposite side, this will be my adjacent side, and this will be my hypotenuse, right? So, as long as we have some sort of triangle to look at that represents theta then we can figure out the answer. Now, where do we get this triangle from? We're going to get it from what's inside of this parenthesis right over here. So, what's inside of the parentheses tell us that cotangent inverse of 2x plus 1 is equal to theta. Now, cotangent inverse means that cotangent of theta is equal to 2x plus 1. So, saying this is pretty much saying the same thing if square root of x equals 2, well that means x is equal to 2 squared. That's what it means. So, we use the definition of cotangent inverse to learn about theta. Now, cotangent on the other hand is what? It's uh-- cotangent is adjacent over the opposite, so now as long as we throw this 2x plus 1 over some denominator 1 so we don't ruin this expression, now we can actually label this triangle as having adjacent side of 2x plus 1 and having the opposite side equal to 1 and I think I made a terrible mistake here by calling sin as opposite of adjacent; it's opposite for hypotenuse, right? So, how do we know that the hypotenuse is equal to? Hypotenuse will be equal by Pythagorean theorem, the square root of the sum of the squares of each of those sides. So, the hypotenuse will be equal to 1 squared plus 2x plus 1 squared. That's it. So, now we have our opposite one and we have our hypotenuse which is what we found it is and let me just kind of move it a little bit to the right, and I'm just going to use simple math here to simplify this. It's going to be 4x squared plus 4x plus 1 and plus another 1 plus 2. So, this is the answer, so 21 over square root of 4x squared plus 4x plus 2. And that's correct. So, all of these expressions are of the same type and if you look at the solution, it's pretty much what I just showed you. let's do another one of this. As you can tell-- as you can tell. As you will be able to tell, I've found [inaudible] this one, all these problems are pretty much the same. So, we're looking for cotangent of cosine inverse of 4x minus 2. If I call this theta, then I'm just simply looking for the cotangent of theta. Cotangent is adjacent over opposite, so I just need to figure out some triangle with angle theta in which I know that adjacent and the opposite, so this is my opposite; this is my opposite, this is my adjacent, and this is my hypotenuse, and now how do I know that triangle exists? I can use the fact that theta is equal to cosine inverse point, in other words, cosine of theta is equal to 4x minus 2. Now, I make a fraction out of it and I use the fact that cosine is adjacent over hypotenuse to label this triangle. My hypotenuse is 1, my adjacent side is 4x minus 2, and therefore, my opposite side-- my opposite side will therefore be the square root of 1 minus 4x minus 2 squared. And we can just look at that as is. So, the final answer is adjacent, which is 4x minus 2, divided by the opposite which is square root of 1 minus 4x minus 2 squared. I hope you can read my handwriting or mouse writing I guess. In this case, hopefully you get the [inaudible] impression and the impression, the [inaudible] impression here is that all of these problems are the same; they all fallow these same steps over and over. So, probably two more. Alright, well look it does look the same as the previous one, so let's do it again. But this time I encourage you to, you know, just maybe pause the video before you see what I do and I have tried to predict my next move. So, I'm going to try to be very efficient here. So, we're looking for sin of theta. Sin of theta is opposite over hypotenuse. So, we need some-- we need a triangle. This is theta. This is opposite. This is adjacent. This is hypotenuse, but the knowledge of the size of the triangle comes from us knowing that secant of theta is equal to 5x plus 1, throw it over 1 and use the fact that secant is hypotenuse over adjacent. So, my hypotenuse is 5x plus 1, my adjacent is 1, and my opposite is therefore square root of 5x plus 1, square root minus 1 and we're going to need to simplify this. We can just write it out like this. Hopefully it's the last problem, because I'm sure you got the idea and my mouse is tired. Hypotenuse is 5x plus 1. Typing is definitely faster than the mouse writing. Square root of 5x plus 1 squared minus 1 divided by 5x plus 1, and of course it's right. Come on Mastery bar. One, no problem. Hopefully this is the last one. Let's do this one. So, we have cosine of cosecant inverse of 3x minus 3 and, again, we're going to call this theta, and we're looking for cosine of theta and cosine of theta is simply adjacent over hypotenuse in some triangle with angle theta and the sides labeled accordingly, and now we're going to use the fact that cosecant theta is equal to 3x minus 3. We're going to turn it into a fraction and use the fact that cosecant is hypotenuse over opposite; we're going to label hypotenuse as 3x minus 3 opposite as 1 and Pythagorean theorem adjacent side will be the square root of 3x minus 3 squared minus 1. So, the final answer is adjacent sides, square root of 3x minus 3 square root minus 1 divided by hypotenuse which is 3x minus 3. My handwriting, mouse writing gets sloppier and sloppier with every problem, so hopefully this is the last one. Doo-doo-too. Minus 1 divided by 3x minus 3. Yeah! No. Yeah, we are done with this homework and the next homework will be from the next lesson. So, again, as always if you have any questions, please do not hesitate to reach out.