WEBVTT
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welcome to our math review video on trigonometry. In
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this video, we'll be talking a lot about right
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triangles, right. Triangles show up all over the
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place in physics, which is why we're gonna have
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a special review video and, in fact, several
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examples on how to use them. Um, we're
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also gonna talk about Trig and metric functions in this
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video. Reintroduce you to those because they can also
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show up, especially when we're talking about any sort
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of periodic or harmonic motion. So think of a
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pendulum swinging back and forth anything like that. So
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let's go ahead and get started. Right. Triangles
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, if you remember, are triangles that have a
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90 degree angle in them. So these triangles air
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particularly helpful because they can have two legs that are
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perpendicular to each other. And this will come up
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when we talk about vectors. So, for example
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, of if you remember from your whatever physics you
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may have taken before, a vector looks something like
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a line with an arrow at the end, it's
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supposed to depict some sort of magnitude that's pointed in
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a particular direction. So, for example, if
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we walk 5 m to the right and then say
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we were to right, turn and walk 5 m
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up Then here if I were to draw a vector
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, showing how we went from the beginning directly to
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the end. You see, I've actually created a
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right triangle and this interaction here is why right triangles
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become so important for us later on. So let's
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consider how all of this is related. So,
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um, let's give some names to our sides here
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. I'm actually going to just name one of the
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angles theta here because the other angle then will necessarily
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have to be 90 minus data. Remember that inside
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a triangle, all of the interior angles must add
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up Thio equal 180 degrees. Since we're requiring that
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one of those angles be 90 that means the other
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two must add up to be 90 degrees. So
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if I say that I know this angle, it
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means the other one has to be 90 minus.
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Whatever that angle happened to be. So looking at
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that, then the first relationship you probably remember is
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the Pythagorean theorem. A squared is equal to B
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squared plus C squared where a is something called the
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high pop news in BNC or what we call the
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legs. So this is a nice relationship for how
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the sides related, and we already looked at how
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the angles air related. What we're really interested,
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though, is. And how can we relate the
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sides to the angles and vice versa. So the
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way we're going to do that is with what are
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called trig and metric functions. So you probably remember
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some of them. For example, we had sign
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of data or sign, which we can write as
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sign of data. And we have cosine, which
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we can write as coasts Fada. And then we
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had tangent, which can be written as 10th data
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. These were are three principal triggered a metric functions
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, and in fact, 10th data is simply sine
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theta over cosign data. So sine and cosine data
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are really the heavy hitters when it comes to trigger
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the metric functions. Um, so looking at these
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, though, how do they relate the angles to
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the sides? Well, it turns out that the
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sign of data is equal to the ratio of the
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opposite side from the angle theta divided by the high
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pot news. So, looking at that, then
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the opposite side would be be so. In this
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case, sign of data is equal to be over
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a cosine theta, on the other hand, is
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equal to the adjacent side to the angle, divided
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by the high pot news, which in this case
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then is going to be equal to see divided by
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a. And looking at this, you can probably
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figure out with a little algebra that since 10th it
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is equal to sign over coastline. That means it's
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equal to the opposite side, divided by the adjacent
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side. Or, in this case, we have
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B oversee. So, um, here's our different
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relationships. Sometimes people use a pneumonic device to remember
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this. They call it so Cartola, which is
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to say sign is equal to the opposite over the
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hypotenuse. The coastline is equal to the adjacent over
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the high pot news, and the tangent is equal
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to the opposite over the adjacent. Um, it's
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not a bad pneumonic device. I definitely recommend that
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you remember it, and knowing these three setups,
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we can see that we have successfully related the sides
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to the interior angles. Thea other thing that we
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might want to remember that this is another. There's
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a couple other relationships here. One in particular that
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shows up is something that's known as the law of
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Cozzens. Okay? And what the law of co
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science does for us. It says if we have
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a generic triangle Um, sorry, we have a
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B C and then the opposing angles air labeled as
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capital a Capital B in capital. See? See
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what I've done here is that the angle opposite to
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the side has the capital version of that name.
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If we do that, then we find that a
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squared is equal to B squared plus C squared minus
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to B C. Cosine of a. Okay,
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so here's another relationship that can end up being used
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. Um, this shows up in rotational motion a
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little bit, though I may or may not explicitly
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use it. Some classes don't worry about it.
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Others do. But there it is for you to
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remember. Some of you may have seen it before
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. Okay, so let's talk a little bit about
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what sine and cosine look like when we plot them
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. Because these are actual functions. They're not just
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those ratios. They're not just static, their actual
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functions that we have So the sign of X,
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where X is our independent variable turns out to look
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like this. It oscillates back and forth. And
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in fact, if I were to extend this axis
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, it would continue to oscillate back and forth and
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it oscillates back and forth over here and again.
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If I were to continue to extend the axis,
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it would continue to follow that pattern forever and ever
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out to infinity. Um, so this sort of
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motion from a function is really helpful. And then
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cosine is extremely similar. Except instead of starting at
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the origin 00 it's going to start at X equals
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zero and y equals one, which is the amplitude
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of these functions. That is to say, sine
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and cosine are never larger than one, and they're
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never smaller than negative one. So negative one is
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there a minimum and one is their maximum. And
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this is really helpful because it means that if we
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were to multiply either of these by some constant,
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say a, then A would become the new amplitude
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of the function f of X in this case.
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Or I could do the same thing with Kassian.
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Notice also that really sine and CoSine are just displaced
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by some amount. If I were to take sign
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and shifted to the left just enough to where that
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peak was, that was on the Y axis,
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then it would look exactly the same as co sign
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, and that's by That's intentional. That's that's a
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very helpful property as well. Um, tangent because
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it's equal to sign over CoSine. We'll also have
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a repetitive motion. But notice that since it's over
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cosine cosine periodically hit zero. What's gonna happen is
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that tangent is gonna have a whole bunch of vertical
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Assam totes that it can't cross. So it looks
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something like this motion. Here we'll go more and
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more into what these different plots mean later on,
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Um, I should note that there are inverse functions
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toe all of these. So Sign has an inverse
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function, which is the Ark sign of X,
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which can also be written as signed to the negative
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one of X. The reason I don't like doing
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that is because sometimes it gets confused with one over
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sign of X, which is actually called the co
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Seacon. And that is not what we're looking for
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. A sign of X I will use it as
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being equal to sign to the negative one of X
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. Similarly, we have co sign of X,
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which has an inverse function that's a coast of X
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or co signed to the negative one of X and
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tangent has a similar inverse function. Tan of X
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is a 10 of X or 10 to the negative
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one of X. And all these really mean is
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that if I take sign of X and then I
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operate on that with the inverse sign, then I'm
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going to get X out of that. They're inverse
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operations. Now, there's one more thing that we
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should take a quick look at just to be prepared
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for when we come across inclined planes. And that
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is, if we have two angles that are arranged
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like this, where the rays of these angles are
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perpendicular to each other, you'll find that these two
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angles must be the same. This comes about because
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of the perpendicular charity here. Okay, because this
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is 90. That means that this is 90 minus
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data. So if this is data 90 minus data
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, this has to be 90 minus data. Those
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with same angle, which means this is data.
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This would be really helpful when we get to inclined
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planes