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18.01 Single Variable Calculus, Fall 2007
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18.01 Single Variable Calculus, Fall 2007
Transcript – Lecture 27
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PROFESSOR: Well, because our subject today is trig integrals substitutions, Professor
Jerison called in his substitute teacher for today. That's me. Professor Miller. And I'm
going to try to tell you about trig substitutions and trig integrals. And I'll be here
tomorrow to do more of the same, as well. So, this is about trigonometry, and
maybe first thing I'll do is remind you of some basic things about trigonometry.
So, if I have a circle, trigonometry is all based on the circle of radius 1, and centered
at the origin. And so if this is an angle of theta, up from the xaxis, then the
coordinates of this point are cosine theta and sine theta. And so that leads right
away to some trig identities, which you know very well. But I'm going to put them up
here because we'll use them over and over again today. Remember the convention
sin^2 theta secretly means (sin theta)^2. It would be more sensible to write a
parenthesis around the sign of theta and then say you square that. But everybody in
the world puts the 2 up there over the sin, and so I'll do that too.
So that follows just because the circle has radius 1. But then there are some other
identities too, which I think you remember. I'll write them down here. Cos 2 theta,
there's this double angle formula that says cos 2 theta = cos ^2 theta  sin ^2 theta.
And there's also the double angle formula for the sin 2 theta. Remember what that
says? 2 sin theta cos theta. I'm going to use these trig identities and I'm going to
use them in a slightly different way. And so I'd like to pay a little more attention to
this one and get a different way of writing this one out. So this is actually the half
angle formula. And that says, I'm going to try to express the cos theta in terms of
the cos 2 theta. So if I know the cos 2 theta, I want to try to express the cos theta in
terms of it. Well, I'll start with a cos 2 theta and play with that.
OK. Well, we know what this is, it's cos ^2 theta  sin ^2 theta. But we also know
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 Fall '08
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 Calculus

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