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Math 103: Precalculus12 chapters | 91 lessons | 10 flashcard sets

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Lesson Transcript

Instructor:
*Jeff Calareso*

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

What happens when we take the sine and cosine wave and transform them? They get bigger, smaller, faster and slower. Find out how in this lesson on sine and cosine wave transformations.

Let's say we go surfing. Actually, no, let's say you go surfing. I'm not going to be shark bait. But you? You're less terrified of the ocean than I am.

So, you're sitting there on your board, waiting for the perfect wave to ride. You notice that the waves move in regular patterns. There's this up and down that is remarkably consistent. This wave motion is what we're going to explore here.

By the way, I see the same thing from the safety of the shore. And as much as I love math, I'm not risking death by shark for it. Good for you, though.

When you're sitting there, those even, steady waves are like a sine or cosine wave. The sine wave is the graph of *y* = sin *x*. It looks like this:

It starts at (0,0) and moves up and down based on the values of sine. The sine of pi/2 is 1, so our graph hits 1 there. The sine of pi is 0, so it's back to 0 there. At 3pi/2, it's -1, then back up to 0 by 2pi, which is one full circle. 2pi is just 360 degrees in radians. That single up and down wave is called a period. Like the waves in the ocean, it keeps going. But we call one full revolution a period.

Then there's the cosine wave. If you've ever surfed, and you know I haven't, you know that the waves are different on different days. Trigonometry is no different. Here's the graph of *y* = cos *x*, or the cosine wave:

When *x* is 0, the cosine is 1, so instead of starting at (0,0) like the sine wave, the cosine wave starts at (1,0). Then it goes down until we hit pi, where the cosine is -1, before going back up to 1 at 2pi.

Again, the period is 2pi. There's also the amplitude. This is the largest distance the wave travels from 0. That's going to be 1 for both the sine and cosine waves. Note that the amplitude is just the distance from 0, not the total vertical distance. It's like the height of a wave from the horizon, as opposed to from the dip it makes.

So, that's *y* = sin *x* and *y* = cos *x*. But, what if our equation is a little more complicated? What if we're on a surfboard - I'm sorry, what if you're on a surfboard - and the sea is angry that day? Bigger waves, right?

We can transform our graph by using this equation: *y* = *A*sin(*Bx* + *C*). This is really just bells and whistles. *y* = sin *x* is the same as this equation, but in this equation, *A* and *B* are 1 and *C* is 0. But what if they're not?

Let's start by changing *A*. What happens with *y* = 2sin *x*? Basically, our *y* values go twice as far as with *y* = sin *x*. So, the *A* impacts the amplitude. *A* for amplitude! Neat. So instead of 1, as with *y* = sin *x*, the amplitude of *y* = 2sin *x* is 2.

Okay, I haven't seen any sharks yet, just ordinary waves and... Wait! What's that? Oh, just some driftwood. Yeah, I'm lots of fun at the beach.

Anyway, let's next manipulate *B*. Let's try *y* = sin 2*x*. What happens to our graph? Well, we still start at (0,0). But now we go up and down in a shorter period. That graph looks like this:

Why? If we plug in values for *x*, we see that instead of hitting 1 at pi/2, we hit it at pi/4. So, our period isn't 2pi, it's pi.

That means that our *B* value affects the period of the graph and that relationship is 2pi/*B*. Whatever our *B* is, just plug that into 2pi/*B*, and you'll have your new period. So, our graph of *y* = sin 2*x* shows a much choppier ocean, with faster waves. This is when it's harder to see sharks. By the way, the sand on the beach, on good old terra firma, is nice and calm today.

How do we remember that *B* changes the period? Hmmm, *B* for period if you sort of mumble 'beriod?' We used the sine wave in these examples, but the facts are no different with the cosine wave.

Finally, what about *C*? Remember, *C* was 0 in *y* = sin *x*. What about *y* = sin(*x* + pi)? This is going to bump our graph forwards or backwards along the *x*-axis. This is called a phase shift. The phase shift can be defined as -*C*/*B*. In this equation, -*C*/*B* is -pi/1, or just -pi.

Now, phase shift is a little trickier to remember, but you can always just plot some points to figure it out. In our example below, what happens when *x* is 0? We still have that pi there, so it's the sine of pi. Where is that on the graph? Where the middle arrow is. So, our wave is crossing the *x*-axis at 0.

But see how it shifted? Our wave shifted back the distance of -*C*/*B*. So, we backed up our waves. You don't usually see waves going backwards, do you? If you do, you're probably surfing the wrong way.

I bet you know what's coming next. Let's see what happens when we put it all together. What is the graph of *y* = (1/2)sin(4*x* - pi)?

Let's use what we know. Our *A* is 1/2. *A* is amplitude. So, our amplitude is 1/2. That means the graph will peak at 1/2.

Our *B* is 4. That's changing our period. A normal period is 2pi, so our period here is 2pi/4, or pi/2.

Finally, our *C* is -pi. And our phase shift is -*C*/*B*. Here, that'll be -(-pi)/4, or +pi/4. So, this time our wave shifts to the right by pi/4.

Put that all together, and you get this:

Is our amplitude 1/2? Check! Is the period pi/2? Check again! And is our phase shift +pi/4? Check!

Let's try one more. What about *y* = 2cos(8*x* + pi/3)?

Again, let's use what we know. Our *A* is 2. *A* for amplitude. So, the amplitude is 2. This is going to be a taller wave that peaks at 2.

Our *B* is 8. That's our period. *B* for 'beriod,' right? Again, a normal period is 2pi, so our period here is 2pi/8, or pi/4. So, we complete one entire period in just pi/4. So far we have a tall, fast wave. I'm glad I'm on shore for this one.

Finally, our *C* is pi/3. We determine the phase shift with -*C*/*B*. Here, that'll be -(pi/3)/8, or -pi/24. So, this time our wave shifts to the left by pi/24.

If we look at this one on a graph, you get this:

Is our amplitude 2? Check! Is the period pi/4? Check, check! And is our phase shift -pi/24? Well, it's a little hard to see here, but it is. So, we did it!

In summary, we started by looking at the graphs of *y* = sin *x* and *y* = cos *x*. These are the sine and cosine waves. We then expanded our equation to *y* = *A*sin(*Bx* + *C*), which involves transformations of the sine wave. With these transformations, *A* directly impacts the amplitude. That's how far the graph goes above the *x*-axis.

*B* affects the period. The normal period is 2pi. We divide 2pi by *B* to get our new period. Then there's *C*. This causes a phase shift. We use -*C*/*B* to determine our phase shift. If -*C*/*B* is negative, then we shift left. If it's positive, then we shift right. Finally, if you do go surfing, keep an eye on those waves. And watch out for sharks!

Once you've completed this lesson, you'll be able to:

- Identify the sine and cosine waves
- Explain the transformations of amplitude, period and phase shift

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Math 103: Precalculus12 chapters | 91 lessons | 10 flashcard sets

- Go to Functions

- Graphing Sine and Cosine 7:50
- Graphing Sine and Cosine Transformations 8:39
- Unit Circle: Memorizing the First Quadrant 5:15
- Using Unit Circles to Relate Right Triangles to Sine & Cosine 5:46
- Special Right Triangles: Types and Properties 6:12
- Law of Sines: Definition and Application 6:04
- Law of Cosines: Definition and Application 8:16
- The Double Angle Formula 9:44
- Converting Between Radians and Degrees 7:15
- How to Solve Trigonometric Equations for X 4:57
- List of the Basic Trig Identities 7:11
- Go to Trigonometry

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