Graphs of the Sine and Cosine Functions - Video Tutorials & Practice Problems

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1

concept

Graph of Sine and Cosine Function

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Hey, everyone. So at this point in the course, we should be very familiar with the sine and cosine trigonometric functions. And we've recently talked about how these trig functions will relate to the unit circle. Now, what we're gonna be talking about in this video is how you can take the sine and cosine functions and graph them on a standard XY graph. Now, this might sound a bit intimidating since we don't know at all what these graphs are going to look like, but don't sweat it because it turns out what we're going to learn in this video is that using the values that were familiar on the unit circle, we can actually graph these in a very straightforward way using these values. So without further ado let's get right into things. Now, I want you to recall how the S is associated with the Y coordinates when you look at a unit circle and the cosine is associated with the X coordinates. Now, what you'll see when looking at these values is these values will repeat as you go around the unit circle. So let's say, for example, we're looking at the sign we can see that our sine value starts at zero and then it goes to one and then to zero, then to negative one and then it starts repeating, it goes back to zero then to one, then zero. And it just repeats as you go around. So you'll notice for the sine and cosine these values repeat around the unit circle. And because of this, you're going to see some interesting behavior in their graphs. Let's say we're looking at the sine graph where Y is equal to the sin of X, we can figure out what this graph looks like by plotting some points. So let's say we want to find the sine of zero. Well, if we're looking at an X value of zero, the sine of zero based on our unit circle is just going to be zero because it's going to be this Y coordinate. So that means we're going to be at a value of zero starting right there on our graph. Now, let's say we're looking at the sine of pi over two. Well, the sine of pi over two according to our unit circle is one, meaning we're going to be up here at pi over two. Now let's say we're looking at the sine of pi. Well, according to our unit circle, the sine of pi is zero, meaning we're back down there on our graph. And based on the unit circle, if we go to three pi over two, the s of three pi over two is negative one. Meaning we're going to be down there on our graph, you can keep plotting these points. So the s of two pi would be a full trip around the circle right back to zero. And then the sine of five pi over two would be a full trip around the circle and then all the way back up to one, meaning that you is going to look something like this. And if I go ahead and connect these points with the smooth curve, you'll notice something kind of interesting happens in our graph, we get this kind of wave that shows up. And that's what happens when you have these repeating values. So both the sine and cosine are going to be graphs that look like waves. Now to prove this to ourselves, let's take a look at the cosine graph as well. So the cosine is going to be associated with the X coordinates. And if we look at the cosine of zero, we can see that that's right here at one for the cosine of pi over two, that's going to be at zero for the cosine of pi that's going to be there at negative one for the cosine of three pi over two, that's going to be back at zero. And then you can keep plotting these points. So for two pi we're going to be back up at one. And then for cosine of five pi over two, we're going to be back at zero, meaning your graph is going to look something like this. Now notice how we get the same kind of wave pattern that we got for the sign except this graph starts a bit differently. Notice how our graph starts at an output of one for the cosine and our graph starts at an output of zero for the sign. But if you look at the unit circle, this should make perfect sense because we can see at zero radiance, our cosine starts at one and our sign starts at zero. So that's why we have the graphs that look like this and then show this waving pattern based on the repeating values that we saw. Now, some other terminology you should be familiar with are the high points and low points of our wave. The high points are known as crests or they're also called peaks. So the peaks of the wave are going to be all these places that we see our high values at our highest values on the wave. And another thing you should be familiar with is the low points which are known as troughs or valleys. The valleys are going to be these lowest points on each of the waves. Now, it turns out these peaks and valleys are not always going to be a positive one and negative one for the sin and cosine because recall that we learned back in algebra that you can actually have transformations on your function. And it turns out that the sine and cosine graphs can also go through transformations. Now, one of the simplest transformations we learned about was something called the vertical shift. And this occurs when you go ahead and take a function and you add some constant K to it which will cause it to vertically shift up or down. Well, it turns out for the sine and cosine graphs, you can also add a constant K which vertically shifts the function. So adding a constant to the sine or cosine will shift the wave up or down in a certain way. Now, if the K value that you have is positive, the graph is going to shift up. Whereas if the K value is negative, the graph will shift down. Now to understand this a bit better. Let's actually try an example where we put both these concepts together. So here we have a situation where we are asked to graph the function Y is equal to the sin of X plus one. Notice we have a sine function we're dealing with and it's been shifted in some kind of way now to solve this problem. What I'm first going to do is try graphing the function Y is equal to the sin of X. So I'm gonna go ahead and ignore this plus one for now. Now we're called for the sin of X. We start at a value of zero at an output of zero. And then we reach a peak at pi over two, then we go back to zero through pi and then we reach a valley at three pi over two, then we go back to zero over two pi and we reach another peak at five pi over two. But how exactly could we graph the function Y is equal to the sin of X plus one? Well, this is what we're ultimately looking for in this problem. And because we have a plus one, this tells us that we're going to have some kind of vertical shift. It's a positive number, meaning it's going to shift up. So all we need to do is take this entire graph and shift it up by one. So this point is going to be at an output of one, this peak would be at an output of two and then this point would be at an output of one and then this valley at negative one would now be at an output of zero. So what we're going to do is trace a curve that looks like this will have a peak at an output of two. Then we'll go down to one and then we'll reach our valley at an output of zero, then we'll go back up to an output of one and then we'll reach a peak at an output of two. So this would be the graph for Y equals the sin of X plus one. And that is the answer to this example. So hopefully this gave you a good understanding of how to graph sine and cosine functions and deal with any kind of vertical shifts if you have them. So I hope you found this video helpful. Thanks for watching.

2

Problem

Problem

Sketch the function $y=\cos\left(x\right)-1$ on the graph below.

A

B

C

D

3

Problem

Problem

Determine the value of $y=\sin\left(-\frac{\pi}{2}\right)+50$ without using a calculator or the unit circle.

A

$y=50$

B

$y=51$

C

$y=49$

D

$y=50+\frac{\sqrt3}{2}$

4

example

Example 1

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2m

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Hey, everyone, let's see how we can solve this example. So in this problem, we're asked to graph the function Y is equal to the sin of X plus three on the graph below. Now, in order to solve this problem, I always like to start off by graphing what I'm already familiar with. And something that I'm familiar with is what the graph Y is equal to the sin of X is going to look like. So I'm going to go ahead and ignore this plus three for now and just graph this sign of X because we can incorporate this plus three later. So we're called that the sign of X graph is a value that starts at the center. And what you can do is going to the right, this is going to be a wave. Now, our wave is going to reach a peak at pi over two. And then we're going to cross down here through PI and then we're going to reach a valley at three pi over two and keep waving as we go to the right now, notice that the peak is at positive one and the valley is at negative one. For the two outputs on the Y axis. Now we can continue this wave going back on our graph as well. So we're going to reach a valley as we get to negative pi over two. And we're gonna come and cross through negative pi we're going to reach a peak at three pi over two and then this graph will keep waving. So this is what the sign of X graph is going to look like. Now, in order to graph this function which is Y is equal to the side of X plus three, what I can recognize is that we have a number being added to our side of X. This is going to cause a shift and because we have a positive number, this graph is going to shift up by three units. So what that means is I can take every point and shift it up by three. So the point that we had that started at the center is now going to be up here at a value of 03, that's going to be our coordinate there. So we're going to be three units up for where our graph starts. Then this peak here is going to be three units up as well, which is going to be at positive four, this zero point that we have at pi this is going to be three units up. So it's going to also be at three. And then this valley that we have at negative one is going to be up three units at positive two. So that's what the graph is going to look like as we go to the right. And this will also be the same as we go to the left. So for negative over two, we're also going to be at positive two for our output at negative pi we're at an output of zero. So our output is going to be up here at three and then at negative three pi over two, we can see that our output is one. So that means our output will be up here at four. So connecting these points, the graph is going to look something like this where we have this wave length behavior that goes to the left and right side of this graph. So this is our sketch for the function Y is equal to the sin of X plus three. And that is the solution to this problem. So hope you found this video helpful. Thanks for watching.

5

concept

Amplitude and Reflection of Sine and Cosine

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5m

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Welcome back everyone. So in the last video, we got introduced to the sine and cosine functions as graphs and we also saw ways that these functions could be transformed by a vertical shift. Now, what we're going to be talking about in this video is new ways to transform the sine and cosine function by using the amplitude or reflection. Now, this might sound a bit complicated but don't sweat it because what we're going to learn in this video is that these types of transformations you can do to the sine and cosine are actually very similar to transformations we've already learned about for functions back in algebra. So you're going to find, I think that a lot of these graphs are actually very straightforward. So without further ado let's get right into this. Now, when it comes to the amplitude, the amplitude is a number and it's a number that affects how tall the peaks of your graph are. Now, I want you to recall that the sine and cosine function are both repeating waves. And if you find the distance from the mid line to the peaks or valleys of that wave that is going to tell you the amplitude. So let's say for example, that we're looking at a sine graph and we take a midline and we draw it right through the middle of this graph. If we draw a midline horizontally through this graph, we can find the amplitude by looking at the distance from this midline to either a peak or a valley to this wave. And we can see that that distance is one unit, meaning the amplitude of the sine function is one. But what would happen if we had a different amplitude or how could we have a different amplitude? Well, let's take a look at a different situation where rather than having a one in front of our sine function, we have two. So to figure out what this looks like, I can plot some points on this graph by taking all the outputs of the sign and multiplying them by two. So we would have two times zero, which is zero, meaning we would start here on our graph. Now, when we get the pi over two on the X axis, we would have two times one, which is two. When we get the pi on the X axis, we would have two times zero, which is again just zero, then we would have two times negative one which is negative two, then we would have two times zero, which is zero. So if you go ahead and draw this graph by connecting these points with a smooth curve, you're going to get a function that looks something like this. And I want you to notice what this function looks like for two times the sin of X notice it's very similar to what we had for the sin of X, except it appears that our graph has been vertically stretched in some kind of way like it's taller. Well, that's the idea of changing the amplitude. It's literally just like a vertical stretch that we learned about back in algebra. And just like we had a number in front of the function, we now have a number in front of the sign. So if you want to, to change the amplitude of your graph, you just need to change this number in front of the sine or cosine function. And that's really the main idea of the amplitude. We can see that we multiplied the sine function by two. So our amplitude for this function was two. And likewise, when we just had a one in front of the sine function, our amplitude was one. Now, another thing that you'll frequently see in this course is situations where you have a negative amplitude and when the amplitude is negative, you're going to get a graph that is reflected over the axis. So to see what this looks like, well, let's plot another graph. So let's say that we have negative sin of X. If we want to find what this looks like, we can take all the outputs for the sin of X and make them negative. Now negative zero would just be zero. Negative one would be look something like that. We would then have negative zero once again, which is just zero. And if we had a negative negative one, that's the same thing as canceling the negative signs giving you a positive one and then you would have negative one times zero, which is again just zero the graph meaning that it's going to look something like this when you connect to these points with the smooth curve. Now notice for the negative sine graph, it literally looks like we took the sine function and just flipped it upside down over the X axis. And that's exactly what happened. And you may recall back in algebra that a negative number in front of the function would flip the function upside down. And that's exactly what happens with the sine and cosine as well. So this is really the main idea behind changing the amplitude or reflecting your graph over the X axis. Now, to make sure that we're really under understanding this, let's actually try an example where we have both a change in the amplitude and a reflection. So here we're asked to graph the function Y is equal to negative three halves times the cosine of X. Now this function right here could also be written as Y equals negative 1.5 times the cosine of X 1.5 is the same thing as three halves. So basically, our amplitude is 1.5. And if I go to our graph, 1.5 would be right between one and two here. And then we would negative 1.5 which is right down there between negative one and negative two. Now we're called for the cosine function. The cosine typically starts at a high value, then goes through pi over two reaches a valley, then goes back up through three pi over two, reaches a peak and then comes back down through five pi over two and keeps waving like that. But we're called that we have a negative sign in front of this function. So what that's going to do is take our gold cosine graph and it's going to flip it over the X axis. So what we're actually going to do for the cosine is we're going to start at a low value. Then we're going to go up through pi over two and we're going to reach a peak at pi. Then we're going to go down through three pi over two, reach a valley at two pi, then we're going to come back up here and touch five pi over two on the x axis and the graph is going to keep waving like that. So this is what the graph will look like and that's the answer to this problem. So hopefully, this gives you a better understanding of how to deal with changes in the amplitude or a reflection for the sin or cosine functions. Hope you found this video helpful. Thanks for watching.

6

Problem

Problem

Determine the value of $y=-2\cdot\sin\left(-\frac{3\pi}{2}\right)+10$ without using a calculator or the unit circle.

A

$y=8$

B

$y=10$

C

$y=-2$

D

$y=12$

7

Problem

Problem

Graph the function $y=-3\cdot\cos\left(x\right)$.

A

B

C

D

8

example

Example 1

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2m

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Hey everyone. So in this example, we are asked to graph this function Y is equal to two times the sin of X minus one. Now, whenever I'm dealing with these types of problems, I like to start these off by solving first what I'm familiar with and graphing it and then building off the graph from there. So what I'm going to do in this problem is I'm going to start by graphing Y is equal to two times the sin of X. So we're going to ignore this negative one for now, but we'll get to that in a moment. So to graph this function, we're called it for the sin of X. What we do is we start here at our graph, we reach a peak at an output of then we dip through pi and then this keeps waving. But we can't actually do that for this graph because we have two here and two is going to change the amplitude. So we're still going to start in an output of zero. But now our peaks are going to reach an output of two and our valleys are going to reach an output of negative two So our graph is going to look something like this. We'll start here at the center of our graph. Then we're going to reach a peak when we get up here to PI over two. So we'll reach our peak right about there. And then we're going to dip through pi on our X axis. Then we're going to go to three pi over two. We're going to reach a valley and then we're gonna keep waving as we go to the right now. What we can do is extend this graph to the left as well. So we're going to reach a valley at negative pi over two, we're going to reach a peak as we go to negative three pi over two and we're gonna keep waiting to the left. So this is what our sine graph is going to look like when we have two times the sin of X. Now the graph Y is equal to two times the sin of X minus one. This minus one is going to cause a vertical shift. Since it's a negative value, we're going to have a vertical shift down by one unit. Now, when we shift down, all of these points are going to be one unit down. So what that means is this peak right here is going to go one unit down. This valley is going to go one unit down. This peak will go one unit down and this value will go one unit down and then the, the center here where we started, this is going to actually start in an output of negative one. So what we can do is we can adjust where the peaks and the valleys are going to be recognizing that the peaks of this graph are actually going to be at positive one when we shift one unit down. And then the valleys of this graph are going to be down here at negative three and go ahead and draw this curve. But we'll start here at negative one. And then we're going to reach our peak right about there. When we get to pi over two, then we'll cross back down to negative one when we get to Pi on the X axis. And then we're going to reach our valley at three pi over two, which is going to be at an output of negative three. And then we're going to keep waving to the right now. Likewise to the left, we're going to go down here and reach a valley at negative pi over two on the X axis. We're going to go back up here when we get to negative pi and we're going to reach our peak when we get to negative three pi over two. And we're going to keep waving as we go to the left. So this right here is what the graph is going to look like for Y is equal to two times the sin of X minus one, it's going to be this red graph right here and that is the solution to this problem. So I hope you found this video helpful. Thanks for watching.

9

concept

Period of Sine and Cosine Functions

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5m

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Hey, everyone. So up to this point, we've been talking a lot about the sine and cosine functions as well as what their graphs look like. And we've been talking about various ways that these graphs can be transformed. Recall in a previous video, we talked about the amplitude, which is a number that tells you how tall or short your graph is. Now. What we're gonna be talking about in this video is the period. And this might sound a bit complicated just because we haven't really heard of the period before. But it turns out just like how the amplitude tells you how tall your graph is. The period tells you how wide your graph is. And I think you'll find the problems you come across in this course once you learn about the period is actually quite straightforward. So without further ado, let's get right into this. Now, we should be familiar with the graph for the sin of X, the sin of X creates this graph which has this kind of wave pattern. And what we can do to find the period of this graph is to figure out the distance along the X axis that it takes to complete a full wave or cycle. So if we look, we can see that one full cycle has been completed, we go up and then down and then back up again. And to find the distance along the x axis, we can just start at the origin of this weight, we can travel along the x axis till we reach to the end of the cycle. And as you can see this is going to be at two pi, meaning that our period is two pi for this graph. Now, the question becomes what happens if we had a different graph like say we had the sign of two X. Well, let's figure out what this graph is going to look like and see if it has the same period. So I'm going to plot some points on this graph and I'm going to do this by taking all the inputs the X values and multiplying them by two. So we'll have the sine of two times zero, which is just the sign of zero or zero. We'll then have the sign of two times pi over four and two times pi over four is pi over two. And the sine of pi over two is one, giving us a point right there. Now we have the sine of two times pi over two, which comes up to the sine of pi which is just zero. So we're gonna have another point right there. We have the sine of two times three pi over four, which actually turns out to be negative one when you do that. And then we'll have the sign of two times pi the sign of two pi is just zero. So that means that your graph is going to look something like this. Now, I want you to notice something about this graph notice how we got a very similar wave pattern that we got for the other graph. But for this function, it looks like our graph has been horizontally shrunk. And that's the entire idea of changing the period because what happened is this graph has a shorter period. Notice that rather than going all the way over to two pi like we did for this original function, the sign of X, we now only go to J pi for the sine of two X. So we could say that the period for this graph is pi. So notice how having a number inside of the sine function in front of the X change the period of our graph. And this number that modifies the period, we typically will represent this with the letter B. So if you have this number inside the sine or cosine function, it's going to modify or change the period depending on what that number is. Now, if B is a value greater than one, like we saw in this example, we had two, the graph is going to horizontally shrink and we saw that this graph shrink. But if B is a value between zero and one, then the graph is actually going to horizontally stretch. And if you want a straightforward way to calculate how much your graph shrinks or stretches when you have a sin or cosine function, what you can do is use this equation for the period, the period is equal to two pi divided by B. So if we wanted to use this equation for the sin of X, what we can see is that the sin of X does not have a number in front of the X. So that means that B would just be one, this value here. So what our period would be is two pi over B which is one and two pi over one is just two pi which is what we figured out. Now, we can also use this equation on the other situation we had where we had the sign of two X. Now we're called the B is going to be the value in front of the X which is two. So if we're going to calculate the period, we'd have two pi divided by our B value, which is two, the two here are going to cancel leaving us with just pi. So our period is pi which is what we had this situation. So notice how this confirms how wide both of our graphs are. We can see that the period for the sine of two X is all the way over to pi and we can see the period for the sin of X goes all the way over to two pi. Now, to make sure we're fully understanding this concept. Let's see if we can try an example. And in this example, we're asked without graphing to calculate the period of the following functions. Well notice in both these examples, we're dealing with signs and cosines. And so we know that the period is going to be two pi over B. Now we'll go ahead and start with example A and for example, A, I can use this equation to find the period. So the period is going to be two pi over B. Now we're called that B is the number that's in front of the X inside of the trig function. So B is going to be one half. Now we have two pi over one half. And what I can do is take this one half of the nominator and I can flip it and bring it to the numerator. So you can bring this two up to the top, which is going to give me two times two pi divided by one. Now anything divided by one is just going to be that number and two times two is four. So our period is going to be four pi and this right here is the answer for example A but what about, for example B? Well, we can use the same equation So the period is going to be two pi divided by B. Now we have the cosine of four pi X and we're going to have our B value B everything that's in front of the X. So in this case, it's gonna be four pi now four is the same thing as two times two and then we'll still have pi and the reason I'm writing it like this is notice how one of the twos are going to cancel as well as the pies are going to cancel. So all we're going to be left with for the period is 1/2. And so that means that the period, for example, B is equal to one half. So this is how you can solve problems and deal with situations where the period changes for the sine and cosine. So hope you found this video helpful. Thanks for watching and please let me know if you have any questions.

10

Problem

Problem

Given below is the graph of the function $y=\sin\left(bx\right)$. Determine the correct value for b.

A

$b=\pi$

B

$b=2$

C

$b=\frac12$

D

$b=4$

11

Problem

Problem

The Period for the function $y=\cos\left(bx\right)$ is $T=20\pi$. Determine the correct value of b.