WEBVTT
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problem. Number 38. Ask us to solve the
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equation for theta, where data is less than equal
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to two pi, but greater than or equal to
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zero. Now, in the beginning, when you
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look at this equation, it might be a little
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daunting, it seems, because you see there's a
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co sign squared on a coastline. But not to
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worry. There's a really simple trick for solving this
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one once you just take a long look at this
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equation and recognize something. And that is, if
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we move this one over here and said everything equal
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to zero, our equation starts to look like something
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very familiar. A quadratic equation. You know,
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we have that squared term just that regular term and
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then a constant all equal to zero. And we
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can solve this surprisingly just like it were a quadratic
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equation. And to make that seem more visually familiar
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to what you guys have done in the past,
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we're just gonna set co sign data equal to X
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. All right, so once we substitute X,
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everywhere we see a coastline data, we have two
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X squared minus X minus one equals zero, and
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we can factor this really easily. Okay. We
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want to make sure we have a negative one at
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the end, so we can try this. If
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you look here, just a quick check. I'll
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do that in green. We multiply this way.
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Make sure you Foyle correctly to x squared. CL
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s X minus one. That's your Oh, and
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then minus two x can't forget about that beautiful foil
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. And ah, we get to X squared minus
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X minus one. That's the same as the original
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equation, so we know that we've factored it correctly
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. Now, when you have a factor, we
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know this all each of these will equal zero,
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so you can set them equal to zero two X
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plus one equals zero X minus one equals zero.
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So X equals one and X equals negative 1/2.
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Now I'm gonna substitute Cosette. Data back in just
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a wee are right back to our original variables because
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NTT equals negative 1/2 and co st data equals one
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. So we want to find Fayed as on our
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unit circle Over here, where coastline data equals negative
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1/2 and co sign theta equals one. So let's
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find those and put them in green well, the
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unit's circles really awesome because we know there's a select
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few, uh, angle measures where the co signs
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and signs have exact values that are just really nice
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fractions. And we know that one of those co
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signed A values is negative 1/2 and that negative 1/2
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occurs right here corresponds with these two angle measures.
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All right, This one and this one and you
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guys guessed it. That's two pi thirds and four
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pi ther's. All right. So at two by
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thirds and four pi thirds co sign data equals negative
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1/2. Yeah. Now we have to find where
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, um, co sent 80 equals one, and
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that occurs just right here. You could say zero
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, or you could say too high. And so
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, since the state's equal to less than equal to
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and a greater than equal to, we're just gonna
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put them both in her answer, even though they're
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co terminal just for kicks, ing giggles. So
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, in this answer, state equals zero to buy
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thirds or by thirds or to pie. And there
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you have it