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Cosine.

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[MUSIC]

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I'd like to approximate cosine of x when x
is near zero.

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I'm really asking for more than just say,

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an approximation to cosine of 0.12 or
something.

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Right, I want more than just an

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approximation that's good at a single
point.

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This is really what I want to do here.
This is going to be my, my goal.

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It's going to be to find a polynomial,

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and hopefully not a very high degree
polynomial.

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I'll call that polynomial P of x so that
the following happens.

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The distance between P of x and cosine of
x is less than 1

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100th whenever, say, x is between minus 1
and 1.

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How can I get started?
So to do this, we're going to use

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the Taylor series for cosine.

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Taylor series for cosine of x is the sum,
n goes from 0 to infinity, of

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minus 1 to the n over 2 n factorial times
x to the 2 n.

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The question really boils down to trying
to figure out

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how many terms I have to take from the
Taylor series.

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What I'm saying is that cosine of x is
approximately the sum

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n goes from 0 to some number big N.

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Of its Taylor series minus 1 to the n,
over 2 n factorial times x to the 2 n.

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And the issue here is, I need to know
exactly how big that N needs to be.

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And that depends on x.

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I mean, if x is really big, I'm going to
want to take

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more terms from my Taylor series to get a
good approximation.

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But my goal here isn't to get an
approximation

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for cosine that's good everywhere.

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I just want an approximation for cosine
that's good

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on this interval, the interval from minus
one to one.

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And then I've quantified exactly how good
of an approximation I want.

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I want that approximation to be within 1
100th.

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I'll use Taylor's theorem to get some idea
about the size of the remainder.

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Okay.

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So the function that I'm studying here is
cosine, so let's call that f.

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And what does Taylor's theorem say?

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Well, it says that the difference between
cosine and that

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big Nth partial sum of it's Taylor series
around zero, so

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that's what I'm writing out here, the nth
derivative of that

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with 0 divided by n factorial times x to
the n.

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Well this is the remainder term, big R sub
big N of x,

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and what Taylor's theorem tells me, is it
tells me something about big

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R sub N of x.

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Tells me that big R sub N of x is equal to
f the big

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N plus 1th derivative of f, evaluated at
point z divided by big N

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plus 1 factorial times x to the big N plus
1, and z is just

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some number between 0, that's the point
that I'm taking the Taylor series around.

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And x.

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I'm assuming that x is between minus 1 and
1.

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And that assumption is telling me
something about

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the size of at least this term here.

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Right?

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So if x is in the closed interval between
minus 1 and 1, then

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what do I know about x to the big N plus 1
th power?

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Well then I know that the absolute value
of x

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to big N plus 1th power is no bigger than
1.

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And consequently,

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the absolute value of big R sub N of x is
no bigger than the

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absolute value of the N plus first
derivative, evaluated as mystery point z.

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Divided by N plus 1 factorial.

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And I don't have to include this term
because

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in absolute value, this term is no bigger
than 1.

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I also know something about the
derivatives of cosine, right.

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What I know

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is that the n plus first derivative of
cosine at z.

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Well, what happens if I differentiate
cosine a bunch of times?

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I don't know how many times I'm
differentiating it.

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But as I differentiate cosine, all I'm
going to get

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is maybe a plus or minus sine of z, or

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maybe a plus or minus cosine of z,
depending you

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know, what big N plus 1 is actually equal
to.

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But look at these functions, no matter

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what z is, neither of these are larger
than 1 in absolute value.

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So this is telling me that the absolute
value of the big N

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plus 1th derivative of f at the point z is
no bigger than 1.

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We'll use that fact to give a nicer bound
on the remainder.

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So putting together this fact and this
fact, this is what I know.

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I know that the remainder is no bigger,

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in absolute value, than 1 over big N plus
1 factorial.

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And now we're back to that question from
the beginning.

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How big does N have to be to guarantee

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that the remainder is bounded by one 1
100th?

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Let me write that down, alright?
I'm looking for big N, so that 1 over big

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N plus 1 factorial is less than 100th.
Well I know some factorials, right?

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I know that 4 factorial is 24.

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So four isn't a good choice for big N plus
1.

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But if this were five factorial.

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I had a 1 over 5 factorial.
And that's 1 over 120.

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And a 120 if it's less than 100.

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So if big N plus 1 is 5, that's surely
good enough.

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So N equals 4 is good enough.

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But let's think about the Taylor series
for cosine.

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The Taylor

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series for cosine is cosine of x is equal
to 1 minus x squared

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over 2 plus x to the 4th over 24, minus x
to the 6th

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over 6 factorial which is 720.
And then it keeps on going.

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And the thing to notice here is that
there's no x to the 5th term.

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Why is that relevant?
Well, it means that

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if I'm going to sum through the x to the
4th term, if I'm going to set big N equal

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4, I might as well set big N equals 5.
Because that will improve the error bound.

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But it doesn't actually affect the Taylor
series at

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all, because there is no x to the 5th
term.

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So let's just make this N equals 5, and

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that's going to improve my error estimate
somewhat, for free.

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Let's summarize what we've found.

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Okay,

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so what we've got is that cosine of x is 1

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minus x squared over 2 plus x to the 4th
over 24.

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Plus a remainder term, where I'm imagining
that I'm

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summing through the non-existent x to the
5th term, right?

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So I can get away with using this
remainder term, as if I'm summing through

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the x to the 5th term, because there is no
x to the 5th term.

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All right, the next term is actually

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x to the 6th.
All right.

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And then I know something about how big
this remainder term is.

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It's estimated right up here.
So what do I know about r sub five?

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Well, r sub 5 of x is no bigger than 1
over 5 plus 1 factorial.

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That's 1 over 6 factorial, that's 1 over
720.

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And that was our goal.

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Well, that was our goal!
I wanted to write down a polynomial,

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here it is.
1 minus x squared over 2 plus x to the 4th

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over 24 and this polynomial is
approximately cosine of x with an error no

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worse than 1 over 720 and originally I
just wanted to get within a 100th.

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And I've done way better than that.

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I mean, this polynomial is an awfully good
approximation of

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cosine, as long as x is between minus 1
and 1.

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This is one of the reasons to love Taylor
series.

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We've been doing approximations, but thus
far the

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approximations have really been at a
single point.

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Now, we're getting approximations that are
good at an entire interval.

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That's an extremely important idea, and if
you want to read some

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more about this, the phrase to search for
is uniform convergence.

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[NOISE]