[music] So we know about sine and cosine but what about all those other funny trig
functions? So maybe we already know about sine and
cosine, we saw tangent already too. But tangent could be defined in terms of
sine and cosine as sine theta over cosine theta.
Cosecant is just defined to be 1 over sine theta.
Secant is 1 over cosine theta and cotangent is 1 over tangent theta.
What's so special about those 6 functions? I mean why we'd have a function called
cosecant if its just 1 over sine theta. Why don't we always just try 1 over sine
theta. For that matter why even have a function
called tangent if you can compute tangent in terms of sines and cosines.
It doesn't seeming like we need all of these functions.
To make matters worse, there's even other functions that practically no one knows
about anymore. In addition to these, there's the
haversine function which is just defined to be sine squared of the angle over 2.
I mean you really don't need haversine once you've got sine.
Considering that most of these functions can be defined in terms of other ones, the
reason for studying, you know, these 6 trig functions isn't really mathematical.
You don't really need cosecant if you've got sine.
The reason for emphasizing these 6 trig functions is cultural.
These are the functions that people are likely to see when they open some
technical manual and knowing about these functions can sometimes give you the right
intuition for how to attack a problem. Because sine and cosine are the legs of a
right triangle with angle theta and hypotenuse length 1 just by the
Pythagorean theorem we get this that sine squared plus cosine squared is equal to 1.
Now if you believe this identity and you could divide this identity by cosine
squared you get sine squared over cosine squared plus cosine squared over cosine
squared is 1 over cosine squared. Now because we've got all these other
functions I could rewrite sine squared over cosine squared as tangent squared,
cosine over cosine is 1, and 1 over cosine squared is secant squared.
And this is maybe an example of how knowing about the other functions can be
helpful. Its a little bit easier to internalize
this identity, I mean if you walk around and you happen to notice that secant
squared you can think always going to place over tangent squared plus 1.
Then say trying to internalize this middle identity.
This also a lovely geometric picture that kind a sells you on an idea that these 6
trig functions have some special role. For example, here is a unit circle and
I've drawn an angle of measure theta and we know what some of the lengths in this
picture are. This length across the bottom here is
cosine theta. And this length here, the height of that
right triangle is sine theta. But it turns out that the other 4 trig
functions are also encoded in the lengths of other relevant lines in this diagram.
For instance this line here has length tangent theta and this line between the
point and the y axis has length cotangent theta.
And if you measure on the bottom here the length from the origin to where this
tangent line crosses that has length secant theta.
And, if we measure over here, from the origin to where that tangent line crosses
the y axis, that segment has length cosecant theta.
So when you put all these functions down we've got cosine and sine but we've also
got very visibly cotangent, tangent, secant, and cosecant.
And the secant is measuring you know something that's crossing the circle and
the tangent is really measuring the length of part of this tangent line.
And if we take this picture and we go fader you won't get idea of how all the
trig functions vary together. For instance by, by looking at this
picture with theta moving you can get a sense of how these functions are moving
together. For instance, tangent and secant are
moving together right when tangent is really big, secant is really big.
And conversely when cotangent is really small, cosecant is you know small, close
to 1. They can use some idea as to why tangent
is called tangent. Why is sine called sine?
So sine comes from this Latin word sinus, you know, like the thing in your nose.
It's like a opening, and if like that as an [inaudible] device you can remember
that sine is measuring this side of the right triangle because you can imagine the
right triangle sort of opens up in that side.