Now it’s time to discuss trigonometric functions. Enjoy the video, and if you need a refresher, as always–the transcript is below.
Transcript: Trigonometric Functions
Trigonometric functions. A whole other side of trigonometry opens up when we begin to consider sine and cosine not merely as functions of angle, but as functions of x that we can graph against y. We have to be very careful to distinguish between x and y. The graphing variables on the one hand, from the point x, y where the radius at an angle theta intersects the unit circle.
So we have to be really careful about this distinction, in order to understand this.
First of all, let’s think about how the values of sine and cosine change once around the unit circle. So here we are in the unit circle. In the first quadrant, as theta increases from 0 to 2 pi, so of course as we are going up and around the circle–the x is decreasing, the y is increasing. This means that sine increases from 0 to 1 and cosine decreases from 1 to 0.
In the second quadrant, as theta increases from pi over 2 to pi, well now. Notice that if you imagined a tangent vector, it would be pointed down to the third quadrant. And so both x and y are decreasing. Sine decreases from 1 to 0 and cosine decreases from 0 to -1. So this is the only instance in which both the sine and the cosine are decreasing at once.
In the third quadrant, of course, they are both negative in the third quadrant. But what happens in the third quadrant from pi to 3 pi over 2, is that sine continues to decrease. It decreases from 0 to negative 1, we actually go down to the bottom of the unit circle but cosine now starts to increase. Cosine had already reached it’s minimum value at pi, so now it’s increasing from -1 to 0. It’s still negative but increasing.
And in the fourth quadrant, sin increases from -1 to 0 and cosine increases from 0 to positive 1. So both of them are increasing in the fourth quadrant. When we graph y = sin(x) and y=cos(x), the angle becomes the horizontal axis, and the output becomes the vertical axis. And so these are the graphs. The graph of sin(x) which starts at 0 increases as we move towards pi over 2, decreases and equals 0 again at pi, then becomes negative as we move from pi to 3pi over 2, and then comes back up to 0 at 2pi.
Cosine starts at its maximum, starts at 1, decreases to 0 at pi over 2, goes down to negative 1 at pi. Curves back up to zero at 3pi over 2, and then increases to its maximum again at 2pi, very beautiful, elegant curves. Both graphs are really the same shape, simply shifted from each other. This shape is called the sinusoidal curve, and it is the shape of many waves.
It’s a very important shape in mathematics. Notice that both graphs repeat the same pattern like wallpaper. This is because when the radius turns 2pi, all the way around the circle, the values go through all the same cycles again. So you can go around and around the circle. You’ll just repeat those same values, which is why the pattern repeats like wallpaper.
This distance on the x-axis pi over 2, is known as the period, the distance at which all the values repeat. So in other words, the sin(x) = sin(x) + pi over 2 for any value of x, similarly for cos(x). The standard function y = sin(x) and y = cos(x) each have a period of 2pi in a range given by y is greater than or equal to negative 1, and less than or equal to 1.
Transformations of Functions
Now the test may ask about these standard functions, but it is much more interested in transformations of these functions. So in other words, when we start multiplying and adding a bunch of different values in the equation. Both sine and cosine will respond to the transformations in the same way. So here I will just talk about the transformations of y = sin(x).
Everything I say is going to be true for cosine(x) as well. First, I will distinguish between operations on the outside and operations on the inside. On the outside, so that would be for example, multiplying on the outside. When we multiply the entire sine function on the outside, or outside the parenthesis, we’ll more adding d.
Those are the outside operations. These affect the graph vertically.
By contrast, the inside operations, these are the things inside of the parentheses that were feeding into the sine function. The b times the x, and then adding something into the x before it goes into the sine, these affect the graph horizontally.
Very important outside operations affect the graph vertically, inside operations affect the graph horizontally. Also, when we multiply or divide, that tends to scale the graph. In other words, we either stretch it and make it bigger in one direction or we shrink it and make it smaller in one direction.
So for example, in all four of these drawings, the green M is the same size.
Stretching and Contracting of Waves
In the first one, the purple M is a vertically stretched version. Now in the second one, it’s a vertically shrunk one. We’ve made it smaller in the vertical direction, even though in both of those it stays in the same place horizontally. In the third one, we stretch it horizontally, everything stays in the same place vertically, but it’s gotten wider.
And then the fourth one we’ve contracted it or shrunk it horizontally, whereas again it remains unchanged in the vertical direction. When we add or subtract, that tends to move the unchanged shape in one direction or another. So again here, we have the original green M, and then we just see it moved in four different directions.
Multiplication Outside — Vertical Change
First of all, let’s talk about multiplication outside.
So this is going to be a vertical change, because it’s an outside change and it’s going to scale. It’s not going to shift it, it’s going to scale it and change the size. If a is greater then 1, then the graph expands vertically, it vertically stretches.
And so, here’s an example, in all of these, the green graph will be the ordinary y = sin(x) and the purple graph will be the expanded graph, will be the transformed graph.
So here, y = 3sin(x) is much taller, goes up much higher goes down much lower. Notice that it intersects the x-axis at exactly the same place. Horizontally, it hasn’t changed, it’s only changed vertically. If a is a number between 0 and then 1, a fraction, then the graph contracts vertically, it shrinks vertically.
So again, the unchanged graph is the green, and here we have a purple graph which goes up to a much lower peak and then down to a much lower trough. So it’s much closer to being flat than the original graph. If we add on the outside, that just shifts it. If we add a positive number, the graph unchanged in shape, moves up. So again, the green is the unchanged and the purple, we’ve just shift the graph up one unit, but it’s exactly the same shape.
And everything horizontally stays in the same place. If what we add is a negative number, in other words, we subtract on the outside, then the graph unchanged in shape moves down. And so here we have the original graph and then Y equals sin minus 2, the purple graph which has been shifted into the negative region below the x-axis.
Multiplication Inside — Horizontal Change
Now, we look at multiplication inside, this is a horizontal change.
If b is greater than 1, then the graph horizontally contracts. And this could be a little bit anti-intuitive because you think that multiplying by a larger number would make things larger. You’ll find that everything that happens on the inside is anti-intuitive, it’s the opposite of the way you think it would work. So here, we have the graph of y = sin(x), and then the graph of sin(2x).
And 2x, we see, it’s been horizontally shrunk, it has the same heights, the same peaks and trough heights. But now it has been shrunk, so repeats much more frequently, it becomes a higher frequency wave. In the period of this one is just pi. Now, if we give a fraction we multiply by a fraction inside then the graph is horizontally expands.
So here we have the graph y=sin(x), and the graph y=sin(x/2). And so this one has horizontally expanded, it stretched out, and this actually has a period of 4pi. We don’t even show the full period of it, it would go off the screen because it is a very very wide period.
Changing the value of b changes the period. An ordinary y = sin x has a period of just 2pi, if we have y = sin(bx), this is a period of 2pi divided by the absolute value of b. That’s a handy formula to know. If b is greater than 1, the period gets smaller, and if b is a fraction between 0 and 1, then the period gets bigger.
Avoid the Trap
And again, this is anti-intuitive, it’s very good to be careful about this, because the trap would be to think well b is bigger period gets bigger. Don’t think that way, you have to be very careful when we’re doing operations on the inside. Addition on the inside. When we add on the inside, then the graph unchanged in shape, moves to the left.
Yes, It’s Anti-Intuitive
In other words when we add moves in the negative direction, again anti-intuitive. So here we have the original sine curve, and then we’ve just shifted it to the left in that graph of y=sin(x+pi/6). If we have a negative inside, if we subtract inside, then the shape unchanged moves to the right. So adding a positive number moves you to the left.
And subtracting a positive number moves you to the right. So here, we have the original graph again and the graph of y = sin(x)- 2pi/3). And so whole graph has shifted over to the right. In the big world of mathematics, and for example, even in your own math class, you might be expected to deal with a function in which a, b, c, and d all change.
All four of those change at once. So we’re doing a vertical shift, a vertical stretch, a horizontal shift,and a horizontal stretch all at once. As a general rule, that’s a little more than the a,c,d is gonna ask you. The test tends not to give more than one transformation in each direction. At most, it tends to have just one operation on the inside, and just one on the outside. That enormously simplifies things.
Practice Problem #1
Here is the practice problem, pause the video and then we’ll talk about this.
Okay, trigonometric functions with the equation y = a*cos(bx). Where a and b are real numbers, is graphed in the standard x, y plane. Which of the following is the period of the function, which one is it? Well, remember period can be defined in a few different ways. We can define where it intersects the x-axis, and then it would have to go through an upper loop and a lower loop, or a lower loop and a upper loop.
So for example, from here down and then up to here, that would be a period. Of course, those are in between, it’s a bit hard to tell exactly what the values are there. Another way to find period is peak to peak, another way to find it is trough to trough. Well with the cosine, peak to peak is very convenient because one peak is right here at zero.
The next peak is right here, at 5pi, and so that has to be the period right there, 5pi. The answer is B.
Practice Problem #2
Here’s another practice problem, pause the video and then we’ll talk about this.
Okay, the functions y = sin(x), and y = sin(ax) + b, for constants a and b, are graphs in the standard (x, y) coordinate plane below.
Which of the following statements about a and be is true, which statement is it? Well, we see the purple one, the purple graph, that’s the ordinary y = sin(x) graph. So we have to figure out that blue one. Well, one thing that’s true about that blue one, is that it’s been shifted down.
It’s no longer centered on a mid-line that’s up the x axis, the mid-line of it is a horizontal line at x equals negative 2. So it’s been shifted down, so the b is definitely negative. On the outside, we have added a negative number. So we can eliminate A and C right away. That means that C or D, A and B are eliminated, C or D could possibly be the answer.
Now, we are to think about what’s going on horizontally. Well, clearly it’s been stretched out. Here we have to be very careful, we’re stretching it out, which means that we’re not multiplying by a larger number. Multiplying by a larger number would actually shrink the period, to stretch out the period, we have to multiply by a fraction less than 1.
And so in fact, C is wrong and D is the correct answer.
In summary, know and recognize the shapes of the basic sine and cosine graphs. Be able to read the period of a transformed sine function from the graph, and understand the rules of transformation. How to change, how changes to the algebraic formula for the function affect the position and the appearance on the graph.