18-01F07-L28 - 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 28 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: So we're going to continue to talk about trig integrals and trig substitutions. This is maybe the most technical part of this course, which maybe is why professor Jerison decided to just take a leave, go AWOL just now and let me take over for him. But I'll do my best to help you learn this technique and it'll be useful for you. So we've talked about trig integrals involving sines and cosines yesterday. There's another whole world out there that involves these other trig polynomials. Trig functions. Secant and tangent. Let me just make a little table to remind you what they are. Because I have trouble remembering myself, so I enjoy the opportunity to go back to remind myself of this stuff. Let's see. The secant is one over one of those things, which one is it? It's weird, it's 1 / cos. And the cosecant = 1 / sin. Of course the tangent, we know. It's the sin / cos and the cotangent is the other way around. So when you put a co in front of it, it exchanges sine and cosine. Well, I have a few identities involving tangent and secant up there, in that little prepared blackboard up above. Maybe I'll just go through and check that out to make sure that we're all on the same page with them. So I'm going to claim that there's this trig identity at the top. Sec^2 = 1 + tangent. So let's just check that out. So the sec = 1 / cos, so sec ^2 = 1 / cos ^2. And then whenever you see a 1 in trigonometry, you'll always have the option of writing as cos ^2 + sin^2. And if I do that, then I can divide the cos ^2 into that first term. And I get 1 + sin ^2 / cos ^2. Which is the tan ^2. So there you go. That checks the first one. That's the main trig identity that's going to be behind what I talk about today. That's the trigonometry identity part. How about this piece of calculus. Can we calculate what the derivative of the tan x is. Actually, I'm going to do that on this board. So the tangent of x = sin x / cos x. So I think I was with you when we learned about
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18-01F07-L28 - MIT OpenCourseWare http/ocw.mit.edu 18.01...

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