WEBVTT
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all right. Everybody's favorite game solving trig and metric
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equations. And now one thing to kind of notice
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is that we have to put this restriction here.
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That data lies between zero and two pi. Now
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, it might have been something else, but if
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we don't put it, restriction triggered. A metric
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equations like this actually have infinitely many solutions because they're
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what's called periodic. So if I just keep adding
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to pi to my solution, I'm just going to
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get a new, unique solution. But that's why
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we're truncating the angles to be just in one rotation
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as we go around the unit circle from zero all
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the way to to buy where we started. Okay
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, So if we want toe solve this equation,
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we just want to isolate data. And so notice
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that when we write sine squared data like this,
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what this really means is it sign of data all
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square and now we just kind of want to use
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order of operations to reverse this process. So this
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is something squared. So let's take the square root
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now. I'm taking the square root of something squared
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. This is very important to remember, and you're
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seeing why this is important because we're going to see
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things like this a lot. This is not signed
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data. It's actually absolute value of science data.
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So I have something squared and I take the square
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root. It's just absolute value. That's something.
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The square root of 3/4 is really just the square
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root of 3/4, and I can write that is
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the square to three over the square root of two
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, using properties of exponential. Sorry over the square
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to four, which is to since it's just square
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to three, divided by square to four, but
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square to force to. And so now remember,
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with absolute value, I have to remember think about
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two cases. Either sine theta is squared three over
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to or negative sign data is squared three over to
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Okay, so then we just got to go back
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to the unit. Circle the common values. So
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when is sign equal to square toe three over to
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or when it's signed, data equal to negative square
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to three over to Well, if I just look
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at my table of values, it's pretty easy.
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You have data can be pi over three. It's
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gonna give me Route three over to We can have
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25 or three. That's also going to give me
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route three over to or on the lower half of
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the unit circle. I have four pi over three
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that's going to give me a Y value of negative
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Route three over to and then finally, we have
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55 or three. That's also going to give me
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a value of negative. You're in three over to
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So we actually have four solutions to this equation.
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And now notice that all of these solutions come from
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just going around the circle one time. In other
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words, these angles air between zero and two.
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Fine. So here we have another equation involving coastline
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data, and we're trying to solve data between zero
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and two pi. So the goal is to isolate
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data or, in other words, isolate coastline data
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. But here we actually have a problem because this
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is cosine tooth data and this is cosine theta.
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We really don't have a way to combine does two
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quantities. But this is what we can dio.
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We can actually use a trick identity. So cosign
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tooth data is really the same thing as cosine of
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theater plus data. But if I used the cosign
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, some formula, the angle edition formula, this
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is coastline Square data minus sine squared data. So
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really, this equation it's cousin square data minus sine
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squared. Data plus goes on. Data equals zero
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. But I can go a step further. I
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can write sine squared data. Well, really negative
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sine squared data as follows one. See cosine squared
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. Data minus one. And this is coming from
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the Pythagorean identity. I'm just rearranging this equation.
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Sine squared data plus coastline squared data equals one.
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I just subtracted sine squared and then subtracted one so
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I can replace negative sine squared data with cosign squared
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data minus one. So I have coastline square data
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. Plus, because I absorb this minus cosine squared
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data minus one plus because I'm data or, in
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other words, to co sign squared Data plus co
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. Santa minus one equals zero. Okay, so
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why did I do all that work? Is this
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really so much better then when I started with Well
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, yes, because let's say that why is equal
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to cosign data or let's let's do X. Yeah
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, race that. So we'll have excess coastline data
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. I could have done why, But I just
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wanted to use X. So this is really,
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like two X squared plus X minus one. But
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this is a quadratic equation, and actually in factors
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, this is two X minus one and then X
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plus one. You can verify that when I foiled
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cell and get the same thing. Okay, but
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that means that to cosign data. If I plug
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back in cosign for X minus one times co signed
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data plus one equals zero. So I have a
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product of two factors equaling zero. I can use
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the zero product property and conclude that either to cosine
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theta minus one is zero or co sign data plus
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one zero. But this just means cosign data is
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one half, and this just means cosign data is
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negative one. So I'm looking for values of data
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between zero and two pi, for which cosign data
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is either one half or negative one. But we
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could just look at the unit circle, enlist those
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out. We have pi over three that's going to
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give me one half. We have pie that's going
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to give me negative one, and then finally,
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we also have five pi over three. That's again
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going to give me an X coordinate of what happened
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. So here are my solutions to this equation between
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zero and two by