18-01F07-L27

18-01F07-L27 - MIT OpenCourseWare http:/ocw.mit.edu 18.01...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus, Fall 2007 Please use the following citation format: David Jerison, 18.01 Single Variable Calculus, Fall 2007 . (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons Attribution-Noncommercial-Share Alike. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms
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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus, Fall 2007 Transcript – Lecture 27 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation, or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. 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 x-axis, 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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18-01F07-L27 - MIT OpenCourseWare http:/ocw.mit.edu 18.01...

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