From: grant@delvalle.ocf.llnl.gov (Chuck Grant)
Newsgroups: comp.graphics,comp.graphics.algorithms,sci.math
Subject: Re: Trigonometry Question
Date: 4 Mar 1994 00:00:58 GMT
Organization: Lawrence Livermore National Laboratory
In article <1994Mar2.144528.12175@leeds.ac.uk> eclrh@sun.leeds.ac.uk (Robert Hill) writes:
>In article <2l0dse$no7@bigblue.oit.unc.edu>, brandon@uncvx2.oit.unc.edu (Brandon Van Every) writes:
>>
>> : >Geez, all this is from far too long ago ...
>> : >But if I remember correctly, arccos(x) = 1/cos(x) and arcsin(x) = 1/sin(x).
>>
>> : Hmmm... good try, but wrong memory cell.
>> : secant(x) = 1/sin(x), cosecant(x) = 1/cos(x)
>> : or maybe vice-versa. I'm doing this from old memory too.
>>
>> Hmmm... good try, but dyslexic memory cell. :-)
>> The nomenclature is back-asswards:
>> secant(x) = 1/cos(x), cosecant(x) = 1/sin(x)
>
>The nomenclature is such that functions whose names begin "co-"
>are decreasing on [0, pi/2].
>
>Robert Hill
>
>"Though all my wares be trash, my heart is true."
> - John Dowland, Fine Knacks for Ladies (1600)
With regard to the names of the trig functions, I find this diagram
very helpful. I actually never saw this diagram in trig class. I
wonder why it isn't taught in high school. Did anyone here learn this
diagram in your high school trig class?
I hope the ascii art is good enough to get the message across.
Chuck Grant
A \
| \
| \
| \
| \
B |_______ \ E
| ------__\
C |---------------/ \ \
| /| \ \
| / | \ \
| / | \ \
| / | \ \
| / | \ \
| / | \ \
| / | | \
| / | \ \
| / | | \
| / | \ \
| / | | \
| / | \ \
| / | | \
| / | | \
|/ THETA | | \
D --------------------------------------------------------------------
F G H
Assume BEG is a 90 degree arc of a circle with the center at D,
Theta is angle FDE, AD is perpenducular to CE and DH, DH is perpendicular
to FE, AH is tangent to the arc and thus perpendicular to DE, and the radius
of the arc is of length 1: then:
DE = DG = DE = 1
FE = DC = sine(theta)
DF = CE = cosine(theta)
EH = tangent(theta)
AE = cotangent(theta)
DH = secant(theta)
AD = cosecant(theta)
FG = versine(theta) = 1 - cosine(theta)
FG = coversine(theta) = 1 - sine(theta)
GH = exsecant(theta) = secant(theta) - 1
AB = coexsecant(theta) = 1 - cosecant(theta)
haversine(theta) = versine(theta)/2
hacoversine(theta) (cohaversine?) = coversine(theta)/2
or nicer to look at, the labeled diagram (except for the vertical unit, and
cosecant) :
\
| \
| \
coexsecant | \ cotangent
| \
|_______ \
coversine | ------__\
|---------------/ \ \
| /| \ \
| / | \ \
| / | \ \
| / | \ \
| / | \ \
| / | \ \ tangent
| one/ | | \
| / | \ \
sine | / |sine | \
| / | \ \
| / | | \
| / | \ \
| / | | \
| / | | \
|/ THETA | | \
--------------------------------------------------------------------
cosine versine exsecant
|<---------- one ------------->|
|<-------------------- secant ------------------------------------>|
-------------------------------------------------------------------------------
Charles W. Grant, Ph.D. Chuck-Grant@LLNL.gov
Lawrence Livermore national Laboratory