0:00

[music]. Once, we believe that sine and cosine are

Â important functions, they're all about the connection between angles and lengths.

Â Well, then we want to apply our usual calculus trick.

Â What happens if I wiggle the input to sine and cosine?

Â You might think about trigonometry as being something about right triangles.

Â But, here, I've drawn a picture of a circle and you can rephrase all this stuff

Â in terms of just geometry of the unit circle.

Â So this is the unit circle. The length of this radius is 1 and I've

Â got a right triangle here and this angle is theta.

Â That means this base here has like cosine theta and the height of this triangle is

Â sine theta. The coordinates then, or this point on the

Â unit circle are cosine theta sine theta. Now, let's make that angle just a little

Â bit bigger. Let's suppose that I make this angle just

Â a big bigger, and instead of thinking about theta, I'm thinking about an angle

Â of measure theta plus h and that means this angle in between has measure h.

Â We'll do a little bit of geometry to figure out how far that point moved when I

Â wiggled from theta to theta plus h. So let's think about this point here and

Â let's draw the tangent line to the circle at that particular point.

Â Well, I'm going to draw a little tiny triangle, which is heading in the

Â direction of that of that tangent line. And, so that it's tangent to the circle,

Â the hypotenuse of that little tiny right triangle here in red will be perpendicular

Â to this line here. I can actually determine the angles in

Â that little, tiny right triangle. So we go this big right triangle down

Â here, and just because its a triangle, this angle theta plus this angle in here

Â plus this right angle have to add up to 180 degrees.

Â But I've also got a straight line here and that means that this top angle plus this

Â right angle plus this mystery angle must also add up to 180 degrees.

Â Well, this plus this plus this is 180 degrees.

Â And this plus this, same right angle, plus this mystery angle add up to 180 degrees.

Â That means that, that mystery angle and that little tiny right triangle must also

Â be theta. I'd also like to know the length of the

Â hypotenuse of the little tiny triangle. Radians save the day.

Â How so? Well, I want to know the length of this

Â little piece of arc. What else do I know?

Â I know this is a unit circle. And I know this angle here is h in

Â radians. And the definition of radiance means that

Â this little length of arc here has length h.

Â Now, let's put my tiny right triangle back there.

Â I'm going to have the hypotenuse of that little right triangle also be h.

Â It's going to be close enough, right, because this little piece of curved arch

Â and this straight line are awfully close. Now that I know the length of the

Â hypotenuse and the angles in that right triangle I can use sine and cosine to

Â determine the side lengths of the other two sides.

Â So I've got a little tiny right triangle, hypotenuse h, that angle there is theta,

Â and that means that this vertical distance here is h times cosine theta.

Â And the horizontal distance of that little tiny red triangle is h sine theta.

Â Okay. So how much did wiggling from theta to

Â theta plus h move the point around the circle?

Â So the original point here from angle theta had coordinates cosine theta sine

Â theta. And the point up here which I got when I

Â wiggled theta up to theta plus h, that point has coordinates cosine theta plus h

Â sine theta plus h. So how much did wiggling from theta to

Â theta plus h move the point? Well, if I use this little right triangle

Â as the approximation, sine theta increased by about H cosign data, and cosign data

Â decreased by about h sine theta. We're now in a position to make a claim

Â about the derivative. In other words, from the picture, we

Â learned that sine theta plus h is about sine theta plus h cosine theta.

Â And cosine theta plus h is about cosine theta minus h sine theta.

Â And as a result, we can say something now about the derivatives.

Â How does changing theta affect sine? Well, about a factor of cosine theta

Â compared to the input. So the derivative of sine is cosine theta.

Â And how does changing theta affect cosine? Well, about a factor of negative sine.

Â So the derivative of cosine is negative sine.

Â Maybe you don't find all this geometry convincing.

Â Well, we could go back to the definition of derivative in terms of limits and

Â calculate the derivative of sin directly. So if I want to calculate the derivative

Â of sin using the limit definition of the derivative, well, the derivative of sine

Â would be the limit as h approaches 0 of sine theta plus h minus sine theta over h.

Â The trouble now, is that I've gotta somehow calculate sine theta plus h.

Â How can I do that? Well, you might remember, there's an angle

Â sum formula for sine. Sine of alpha plus beta is sine alpha

Â cosine beta plus cosine alpha sine beta. If I use this, but replace alpha by theta

Â and beta by h, I get this. Sine of theta plus h can be replaced by

Â sine theta cosine h plus cosine theta sine h.

Â So let's do that. So, the derivative is the limit as h

Â approaches zero of, instead of sine theta plus h, it's sine theta times cosine h.

Â This first term, plus cosine theta sine h minus sine theta from up here and this

Â whole thing is divided by h. So, this is sine theta plus h minus sine

Â theta all over h. Now, I can simplify this a bit.

Â I've got a common factor of sine theta, so I can pull that out.

Â This is the limit as h approaches zero, pull out that common factor of sine theta.

Â What's left over is cosine h minus 1 over h plus, I've still got this term here,

Â cosine theta sine h over h. I'll write that as cosine theta times sine

Â h over h. All right.

Â So this limit calculates the derivative of sine.

Â Now, it's written as a limit of a sum, which is a sum of the limits, provided the

Â limits exist. So, I can also note that sine theta and

Â cosine theta are constants. Right, h is the thing that's wiggling, so

Â I can pull those constants out of the limits as well.

Â So what I'm left with is sine theta. Times the limit as h approaches 0 of

Â cosine h minus 1 over h. Plus cosine theta times the limit as h

Â approaches 0 of sine h over h. Now, how do I calculate these limits?

Â Well, we've got to remember way back to when we were calculating limits a long,

Â long time ago. We can calculate these limits by hand,

Â using say, the squeeze theorem. And it happens that this first limit is

Â equal to 0 and the second limit is equal to 1.

Â So I'm left with 0 plus cosine theta. So, this is a limit argument, back from

Â the definition of derivative that the derivative of sine is cosine.

Â So, regardless of whether you think more geometrically or more algebraically, the

Â derivative of sine is cosine and the derivative of cosine is minus sine.

Â Now, once you believe this, there is some sort of weird things you might notice,

Â like what if you differentiate sine a whole bunch of times?

Â So the derivative of sine is cosine and the second derivative of sine, is the

Â derivative of the derivative. It's the derivative of cosine which is

Â minus sine. And the third derivative of sine is the

Â derivative of the second derivative and the derivative of minus sine is minus

Â cosine. And the fourth derivative of sine, well,

Â that's the derivative of the third derivative, which is minus cosine.

Â And the derivative of a cosine is minus sine.

Â So, the derivative of minus cosine is sine.

Â So the fourth derivative of sine is sine. So that's kind of interesting.

Â If you differentiate sine four times, you get back to itself.

Â And we already know a function that if you differentiate just once, it spits out

Â itself again, e to the x is its own derivative.

Â So as sort of a fun challenge, you might try to find a function f, so that if you

Â differentiate it twice, you get back the original function, but if you

Â differentiate it only once, you don't get back the original function.

Â And if you can do this, you can ask the same question for even higher derivatives.

Â We know a function whose fourth derivative is itself, but none of the earlier

Â derivatives are the function again. Can you find a function whose third

Â derivative is itself, but the first and second derivatives aren't the original

Â function? That's a fun little game to play, anyway,

Â it's a little challenge for you to try to find such a function.

Â