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18.01 Single Variable Calculus, Fall 2007
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18.01 Single Variable Calculus, Fall 2007
Transcript – Lecture 28
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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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 Calculus, Trigonometry

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