Re: Sin Cos Tan, why not Sin Sec Tan?

From: Cassandra Thompson (cass.harley_at_bigpond.com)
Date: 07/17/04 

Date: Sat, 17 Jul 2004 00:59:49 GMT

Rouben Rostamian wrote:

> In article <a88af92f.0407160950.2ec6a35f@posting.google.com>,
> David Bandel <dwb1729@yahoo.com> wrote:
>
>>rouben@pc18.math.umbc.edu (Rouben Rostamian) wrote in message news:<cd5vha$6c2$1@pc18.math.umbc.edu>...
>>
>>>In article <4jnJc.1714$K53.1037@news-server.bigpond.net.au>,
>>>Cassandra Thompson <cass.harley@bigpond.com> wrote:
>>>
>>>>Why is cosine a more used function then secant?
>>>
>>>The answer to your question lies in the Pythagorean Theorem which in
>>>terms of the trig functions takes the form:
>>>
>>> sin^2 x + cos^2 x = 1.
>>>
>>>The Pythagorean Theorem manifests itself in many guises in all sorts
>>>of mathematics, from elementary to most advanced. If you want to
>>>replace cos with sec, the formula changes to:
>>>
>>> sin^2 x + 1/sec^2 x = 1.
>>>
>>>which is somewhat more complicated the cosine version. So if there is
>>>a choice between the two forms, people choose to use the first one.
>>
>>that doesn't answer the question
>>
>>why don't we define cos to be the reciprocal of what it used to be.
>>and the same for sec.
>>
>>then we'd have sin^2 + sec^2 = 1..
>>
>>seems stupid that the 3 elementary trig functions aren't the 3
>>elementary words but 2 elementary words and one compound one.
>
>
>
> The words "secant" is not a an arbitrarily made-up word. It has a
> specific meaning to those who are familiar with its Latin etymology.
> Exchanging the definitions of cos and sec is problematic. It's like
> suggesting to let the word "daylight" refer to the darkness after
> midnight.
>
> Here it is, from the American Heritage Dictionary:
>
> se·cant (s¶k²nt, -knt) n.
> Abbr. sec Mathematics
>
> 1. a. A straight line intersecting a curve at two or more
> points. b. The straight line drawn from the center through one end of
> a circular arc and intersecting the tangent to the other end of the
> arc. c. The ratio of the length of this line to the length of the
> radius of the circle. 2. The reciprocal of the cosine of an angle
> in a right triangle.
> [ From Latin sec³ns secant-, present participle of sec³re to cut;
> See sek- in Indo-European Roots.]
>
> I direct your attention to definition (b) given above. That's where
> the trigonometric function secant takes its name.
>
> If you don't see the relation between definition (b) and the geometric
> interpretation of secant, let me know. I'd be glad to explain.
>
Thanks, truthfully I don't see the relation, though I do not doubt its
existance. The replies to this post have quickly become out of my depth.
This is why I have barely said two words.

Having said that I am still working through the trigonometry section of
an excellent website (http://www.themathpage.com). I reread the posts
after I understand something a little better, then the posts make a
little more sense to me....

Thanks,
Cassie

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