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[music] So we know about sine and cosine but what about all those other funny trig

Â functions? So maybe we already know about sine and

Â cosine, we saw tangent already too. But tangent could be defined in terms of

Â sine and cosine as sine theta over cosine theta.

Â Cosecant is just defined to be 1 over sine theta.

Â Secant is 1 over cosine theta and cotangent is 1 over tangent theta.

Â What's so special about those 6 functions? I mean why we'd have a function called

Â cosecant if its just 1 over sine theta. Why don't we always just try 1 over sine

Â theta. For that matter why even have a function

Â called tangent if you can compute tangent in terms of sines and cosines.

Â It doesn't seeming like we need all of these functions.

Â To make matters worse, there's even other functions that practically no one knows

Â about anymore. In addition to these, there's the

Â haversine function which is just defined to be sine squared of the angle over 2.

Â I mean you really don't need haversine once you've got sine.

Â Considering that most of these functions can be defined in terms of other ones, the

Â reason for studying, you know, these 6 trig functions isn't really mathematical.

Â You don't really need cosecant if you've got sine.

Â The reason for emphasizing these 6 trig functions is cultural.

Â These are the functions that people are likely to see when they open some

Â technical manual and knowing about these functions can sometimes give you the right

Â intuition for how to attack a problem. Because sine and cosine are the legs of a

Â right triangle with angle theta and hypotenuse length 1 just by the

Â Pythagorean theorem we get this that sine squared plus cosine squared is equal to 1.

Â Now if you believe this identity and you could divide this identity by cosine

Â squared you get sine squared over cosine squared plus cosine squared over cosine

Â squared is 1 over cosine squared. Now because we've got all these other

Â functions I could rewrite sine squared over cosine squared as tangent squared,

Â cosine over cosine is 1, and 1 over cosine squared is secant squared.

Â And this is maybe an example of how knowing about the other functions can be

Â helpful. Its a little bit easier to internalize

Â this identity, I mean if you walk around and you happen to notice that secant

Â squared you can think always going to place over tangent squared plus 1.

Â Then say trying to internalize this middle identity.

Â This also a lovely geometric picture that kind a sells you on an idea that these 6

Â trig functions have some special role. For example, here is a unit circle and

Â I've drawn an angle of measure theta and we know what some of the lengths in this

Â picture are. This length across the bottom here is

Â cosine theta. And this length here, the height of that

Â right triangle is sine theta. But it turns out that the other 4 trig

Â functions are also encoded in the lengths of other relevant lines in this diagram.

Â For instance this line here has length tangent theta and this line between the

Â point and the y axis has length cotangent theta.

Â And if you measure on the bottom here the length from the origin to where this

Â tangent line crosses that has length secant theta.

Â And, if we measure over here, from the origin to where that tangent line crosses

Â the y axis, that segment has length cosecant theta.

Â So when you put all these functions down we've got cosine and sine but we've also

Â got very visibly cotangent, tangent, secant, and cosecant.

Â And the secant is measuring you know something that's crossing the circle and

Â the tangent is really measuring the length of part of this tangent line.

Â And if we take this picture and we go fader you won't get idea of how all the

Â trig functions vary together. For instance by, by looking at this

Â picture with theta moving you can get a sense of how these functions are moving

Â together. For instance, tangent and secant are

Â moving together right when tangent is really big, secant is really big.

Â And conversely when cotangent is really small, cosecant is you know small, close

Â to 1. They can use some idea as to why tangent

Â is called tangent. Why is sine called sine?

Â So sine comes from this Latin word sinus, you know, like the thing in your nose.

Â It's like a opening, and if like that as an [inaudible] device you can remember

Â that sine is measuring this side of the right triangle because you can imagine the

Â right triangle sort of opens up in that side.

Â