Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. I'm using a web page I found through Google that's going to show us an applet on how to use the unit circle to draw the function Y equals sine X. So here the XS are angle values and the Y is our sine X or the height of the unit circle. So if we looked at this and we started with 00 angle 0, we'd have a height of 0. And if we went slowly, the next angle would give us a little bit higher Y and a little bit higher and a little higher until we got to 90°, which is the highest that our Y is ever going to go. We can see that based off of the unit circle now starting at our 90° going to 180, our YS are decreasing in height on that unit circle at 180. Now the YS are going down in the unit circle or the values are going to be negative. So if we keep going, we get to the smallest our Y is ever going to get at -270 or at 270, which is -1 for our height. If we continue through one full circle at 360, we're back to a height of 0. So this is our graph of our Y equals sine X. Now with this graph we have an amplitude and our amplitude is how high or how low we go and in this case our amplitude is -1 to one. I use brackets because we include the highest and the lowest point. Our period, or how long it took to get a full cycle is 2π. We're going to go into a generic form for a sine graph, and a generic form would be Y equal a sine. The quantity BX plus C plus DA is our amplitude, and actually it's the absolute value of A. It tells us how high or how low we're going to go. 2π A regular. Divided by our B is going to be a new. For this new equation. Negative C / b is going to be a phase shift left or right, and our D is a vertical translation, IE it's moving that whole graph up or down. OK, we're going to look at graphing these on a number line. So when angle was 0°, our cosine value was 1. So at 0° we had one. At 60°, we had a half 90°, we were at 0, 45, we were at Route 2 / 2 which is approximately .7 O 7130 degrees. We were at Route 3 / 2 which is oh about there 8660. Now we know that we have symmetry going on. So between 90° and 180° we have the same values going negative. So we're going to have a curve that then does a reflection and comes back up. If we were thinking about going in negative angle direction, we would have the curve going like. So if we think about the domain and the range, then the domain or what angles can we use is going to be any angle because we have every single value that we could have for a Theta. So the domain is negative Infinity to Infinity. If we look at the range for the cosine Theta, how high and how low can it go? The highest the cosine ever goes is the furthest to the right it went on a unit circle, or the highest it could ever go is 1 here. If you see the height of one on the unit circle, what's the furthest to the left it could ever go would be -1 or on this graph here of Theta and cosine, Theta -1's the lowest it's going to go, so the range would be -1 to one. When looking at the sine graph and the cosine graph on the same axis, we can see that these are both periodic functions with our period repeating every 2π, and they look like they're just a phase shift to being equal to each other. If we thought about our sine graph being a phase shift of our cosine graph, looking at our cosine graph, what would I have to do to get the highest point of the cosine to turn into the highest point of the sine? I need to shift it by π halves, so sine X is really the same thing as cosine of X - π halves because we needed to shift it to the right. Now that gives us a lot of information when we're doing graphs. If we wanted to look at an example, let's say Y equal 6 sine Pi halves X, we'd have an amplitude of 6, our highest and our lowest. It's the number that always is in front of the sign. So the highest is 6 and the lowest is -6. It's the number right in front of the trig function. The Pi halves is going to tell us there's a new. So our original. Is just 2π. Now if I divide it by π halves 2π divided by halves, Pi halves is going to give me a period of four. So my cycle for sine starts at 00. It's going to go up and down, and it's going to be completed in a cycle of four. So half the period would be two, half of that would be 1, and in between 2:00 and 4:00 would be 3. Our highest value would be a six and our lowest value would be a -6. If we wanted to look at one with cosine, we would do the same concept Y equal 9 fifths cosine of π force X. Let's put a plus one in it this time. So our amplitude is going to be 9 fifths. So our high points 9 fifths, our low points 9 fifths. Our period, our regular. Is 2π / π force. So this time our period is going to be 8 and we're going to have a vertical translation of up one. It's always easiest if we do our translation very, very, very last. So an amplitude of 9 fifths and a period of eight cosine graph starts at angle 0 having a one. But now because of the new amplitude, it's going to have 9 fifths and it goes through a full cycle and comes back to a height of 9 fifths. In this case, the period was 8, so halfway between would be 4, half of that would be two, and halfway between 4:00 and 8:00 is going to be 6. So when a high of 9 fifths and a low of -9 fifths, but with this vertical translation of one, it's going to take every value up by 1. So now we're going to be up here somewhere and -9 fifths plus one is going to give us negative 4/5 and 9 fifths plus one is going to give us 14 fifths. So we're going to have something that looks like this was that original graph with everything just plus one. Thank you and have a wonderful day.