Name: Trig limit using pythagorean identity | Limits and continuity | AP Calculus AB | Khan Academy
Uploaded: 2022-07-04T08:00:26.000Z
Duration: 7 min 2 s
Description: 使用畢達哥拉斯恆等式三角限制 |限制和連續性 | AP微積分AB |可汗學院Learn the English Usage: theta,cosine,squared,equal,limit,sine,defined,continuous,pythagorean,doesn exist,expression,rewrite,doesn,exist,exact,denominator,substitute,identity,definition,cross,view,sal,constraint,substitution,drum...

- [Voiceover] Let's see if we can find the limit

as theta approaches zero of one minus cosine theta

And like always, pause the video and see if you

Alright, well our first temptation is to say,

"Well, this is going to be the same thing

which could be used to define a function,

that they'd be continuous if you graph them.

They'd be continuous at theta equals zero,

so the limit is going to be the same thing,

as just evaluating them at theta equals zero

So this is going to be equal to one minus cosine of zero

Now, cosine of zero is one and then one minus one is zero,

and sine of zero is zero, and you square it,

You still got zero and you multiply times two,

So once again, we have that indeterminate form.

when you have zero over zero, doesn't mean to give up,

it doesn't mean that the limit doesn't exist.

there's some other approaches here to work on.

If you got some non-zero number divided by zero,

then you say, okay that limit doesn't exist

and you would say, well you just say it doesn't exist.

But let's see what we can do to maybe, to maybe think

about this expression in a different way.

So, F of X is equal to one minus cosine theta

and let's see if we can rewrite it in some way

that at least the limit as theta approaches zero

isn't going to, we're not gonna get the same

Well, we can, we got some trig functions here,

so maybe we can use some of our trig identities

is that we have the sine squared of theta

and we know from the Pythagorean, Pythagorean Identity

in Trigonometry, it comes straight out of the unit circle

We know that, we know that sine squared theta

plus cosine squared theta is equal to one

over two times one minus cosine squared theta.

This is a one minus cosine squared theta,

until you realize that this could be viewed

if you view this as A squared minus B squared,

we know that this can be factored as A plus B

I could write this as one plus cosine theta

One plus cosine theta times one minus, one minus

in the numerator and I have a one minus cosine theta

We could say, "Two plus two cosine theta."

because F of X, this one right over here,

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Trig limit using pythagorean identity | Limits and continuity | AP Calculus AB | Khan Academy

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