mm-1078
===
Subject: Re: Problems With Public Key Cryptosystems
> [This should follow up the OP, which didnt reach my
newsserver, nor
Google
> (deleted?).]
Yeah, I confess. I deleted it because it broke the secret
that the way to
factor large numbers is to keep a table of every composite in
existence.
At least it didnt explain that you could cut the list in
half by ommiting
factors of 5 and check those by algorithmically checking if
the last digit
is
a 5 or 0. This tends to save considerable hard drive space
and in some
installations allows the machine to operate without
compression enabled.
===
Subject: Re: Garry Denkes Gold & Brass Plates @ Westbury
White Horse Eye
> Therein the reason we need the number 0 rule, and the
number - rule,
> and the number + rule.
> Only if we have all three numbers. Since 0 = + and 0 = -,
theres
> no real need for the other two.
> Repeatedly toss a coin (infintie times)
> What is the probability of HHHHHHHHHHHHHH.... ?
> # desired outcomes 1
> _____________________ = __________________________
> # total outcomes # total outcomes
> as total outcomes -> oo, P(H..) > 0
> It CAN happen, so the probability is NOT 0.
> Herc
The probability is zero. The rule:
probability = (# desired outcomes) / ( # total outcomes)
is valid when the number of outcomes is finite and
all outcomes have the same probability; this applies
to bridge hands, poker hands and lotteries.
In the present case, the number of outcomes is infinite.....
Let q(n) be the probability that the first n coin
tosses are all heads. Then q(n) = 1/(2^n).
If you get heads on every single toss, then for any finite
n, the first n tosses are all heads.
So P(H..) <= q(n) for all n, so that
P(H..) <= 1/(2^n) for all n.
But of course P(H..) >=0.
So 0 <= P(H..) <= 1/(2^n) for all n.
It follows that P(H..) = 0.
David Bernier
P.S. In math, most things are defined, even number systems.
----- You might want that P(H..) > 0; then you could define
probabilities (Herc_probabilities) as you would like.
You might want your probabilities to be
non-standard reals (Herc_reals).
Finally, statements are true or false within
a context that depends on the definitions of
ones terms.
===
Subject: Captain Bat and his psych spew ( Was Re: Nathan
Owen Wiberg -
September 20th 1977)
<darylsk@shaw.ca> stumbled drunkenly into the group after
pissing his
pants and rudely further stunk up the place with the
following mess:
>W I B E R G
>23 9 2 5 18 7 = 64
> I traveled to Prince Albert Saskatchewan and met Nathan, I
think he
said
>that at least one of his siblinks was a sister.
<<<snipped a lot of psycho crap from Shawn pedophile-boy
Kabatoff>
The following (courtesy of Waxy.org) is sort of an unofficial
FAQ
explaining the psychotic nonsense posted to Usenet by Shawn
Daryl
Kabatoff AKA Dar, AKA Probababbilities. And now AKA marcia and
me.
WARNING: Read below before even thinking about responding to
this
twit.
http://www.waxy.org/archive/2002/05/21/dar_kaba.shtml#000643
Usenet has the tendency to provide a public forum for those
who would
normally be scribbling in a closet. For example, take Daryl
Shawn
Kabatoff. For the last few years, hes methodically gathered
statistics from various sources, ranging from local newspaper
obituary pages to the food court of the Saskatoon Midtown
Plaza mall.
With all the raw data hes collected, hes 
attempting to
prove daily
that our full names are in mathematical harmony with our
birthdays.
His rants normally focus on a single individual hes met or
read
about, starting with calculations related to their birthdate
and full
names, blending in whatever other personal information about
their
family members, spouses, birthplace, and career hes been
able to
zealotry, and personal torment. Ive never seen anything 
like
it.
With all the prime numbers, Fibonacci sequences and biblical
references, its like reading the notebooks of Maximillian
Cohen and
John Nash combined. Unsurprisingly, several posts unfold to
reveal a
history of painful mental illness. If you have some time,
take a look.
Ive detailed his posting history and a several sample posts
below.
Usenet Posting History:
January 27, 1999 to July 5, 2000 as Catsco@home.com
December 9, 2000 to May 4, 2001 as s.kabatoff@sk.sympatico.ca
Oct 30, 2001 to Oct 31, 2001 as kabatoff@the.link.ca
January 20, 2002 to April 17, 2002 as s_kabatoff@hotmail.com
(original
posts have been
removed from Google Groups archive)
April 26, 2002 to Present as dar_kabatoff@hotmail.com
Selected Posts:
Tessa Lynne Smith
Dastageer Sakhizai and Helen Smith
Brett David Maki
Andrew Meredith Cotton
Kathryn Lee Hipperson
Amanda Dawn Newton
Mona Marie Etcheverry
Tony Peter Nuspl
Lisa Charlene McMillan
Grant Allyn Wood
Comments
scarier still is that saskatoon is my hometown, though not my
current
residence. and every single place hes mentioned in his 
posts
(most
notably nervous harolds and the roastary) were either 
places
ive
been (as its a small city of 200K) or hangouts, ie. the two
places
mentioned. chances are i could email some friends back home
and find
out if they know of him, they (my friends that is) being of
the
broadway-centred slacker ilk. myself, too, until i got out of
there.
eh, anyways. thought it odd to see all this. midtown mall. i
ate my
meals there, whilst waiting several days in line for star
wars episode
one, at the theatre across the street.
posted by andy raad on May 22, 2002 06:20 PM
Fascinating. Its like hes trying to take 
chaos and bind it
into
whatever rules he can find, religious, logical and otherwise.
Numbers
and math have a reliable pattern, something that can always
be proven
to true or false. People and religion do not. It reminds me
of Darren
Aronofskys movie Pi. Its the story of an 
paraniod genius
who is
trying to find a pattern in Pi. A group that takes interest in
his
work is convinced that the existence of Pi, a number whose
existence
can be proven but no quantified, is proof of the existence of
God.
Kabatoffs hunt for patterns in something as random as name
selection
is a way to reconcile his deeply logical thought process with
his
conßicting religious views.
Exactly. I probably shouldnt have, but I e-mailed Daryl
yesterday,
asking him if hed be willing to create a numerological
analysis for
me. I also asked him if he had seen either Pi or A Beautiful
Mind, and
what he thought of them. If he replies, Ill be sure to post
it.
I baked many pumpkin pies for Shawn (he likes pumpkin pies).
I rubbed
pumpkin pie all over my breasts for him, and my breasts
turned orange.
I am a pumpkin for Shawn.
posted by Trisha Blondie on July 24, 2002 10:41 PM
Um, thats swell. So, youre in love with 
him?
Shawn once went to a funeral for a Jehovah Witness that shot
himself
and the lemon tarts were very bad, they were not only sour
but were
rubbery as well. Shawn said that the guy was some kind of
Jehovah
Witness prophet, he saw in advance that the lemon tarts at
his funeral
were to be very very bad, and so he shot himself. Shawn said
that he
never ate pumpkin pie at a funeral but would like to some
day. Shawn
likes pumpkin pie and so I have been practicing to make very
good
pumpkin pies.
posted by Trisha Blondie on July 25, 2002 02:49 PM
Shawn said that the lemon tarts were sour, bitter and rubbery.
I dont think this guy takes notes. I think he has Total
Recall, and
it has driven him insane...
Oh... I almost forgot... I didnt spend thousands of dollars a
day
tormenting Daryl... We got a deal on tormenting that fiscal
year, it
only came to about 37cents a day....
Mr. Kabatoff attempts to portray himself as a victim, but in
fact he
is a violent predatory pedophile who is well known to his
local law
enforcement. In his post to multiple newsgroups with the
subject
Collecting Mail For The Coming Anti-Christ, he encourages
mothers to
send him photos of their naked daughters. Mr Kabatoff
explains, I
personally did not want photographs being mailed to (the
coming
Ant-Christ) that were of underage children unless the parent
was
signing consent. He is banned from virtually all the shopping
malls
in his community because he stalks young people and sexually
harasses
them. He has an extensive arrest record which includes sexual
molestation charges. Hes been hospitalized in mental
institutions
about his contact with young girls in many posts. Search
newsgroup
archives for posts by him containing the word nubile. As part
of his
harrassment, he provides personal details in a public forum,
such as
the real names of real children, in these and other posts.
About one
wanted her and her sister dead.
He not only curses children and prays for their death in his
posts, he
also enjoys attending the funerals of young people: And so,
since
nubile sweeties are found in greatest abundance at the
funerals of
high school students, then it is the funerals of high school
students
that make the very very best funerals, especially if there is
food...
I stuff my face (and my pockets) with all the good food and
look at
all the pretty nubile sweeties and have the time of my life..
r=&ie=UTF-8&scoring=d&selm=LfXN8.63042%24R53.25142039%
40twister.socal.rr.
com&rnum=1
Many of his posts are sent to alt.teens.advice. However, he
liberally
offensive
missives to countless newsgroups. Some people HAVE problems
and some
folks ARE problems. Dont dismiss Mr. Kabatoff as a harmless
nut. When
he sends these posts to any newgroup, please help by
reporting him to
I knew of him when I was attending the University of
Saskatchewan.
Hed hang out in the Arts computer lab and all 
youd see is
screens of
numbers racing by on his laptop. I have an original copy of
his
Collecting Mail for the Coming Anti-Christ pamphlet, and have
seen
him be hauled away by campus security on more than one
occasion. My
friends and I refer to him as Crazy Number Man.
Ive been posting to (and about) Shawn for over two years
with big
gaps in between. He has seen Pi and didnt like it and 
didnt
think it
resembled him at all. (Wrong, it fits him to a tee) He 
doesnt
have
total recall and has stated that he travels with a lap top to
notate
items. Also, he uses cut n paste a lot if you read all the
way
through his ramblings. He is anti-social as shown by his angry
statements towards those who, by his own admission, have been
kind
(but not kind enough) to him. Still, hes intelligent and
seems to be
able to take a joke on occassion. Thats where I came in.
ALOHA
Reply to group
(Unsolicited e-mail is deleted from the server unread
if it comes from anyone not already in my addressbook.
Ill never even see it)
===
Subject: Re: VOTE on whether 1/oo = 0
> I do have a liberal perspective on the topic also.
> IF you define division by infinity to start 
with
> THEN you could assign that result as 0.
> I advise against doing that.
> Its better to define the elements of the 
extension and the
appropriate
> operations _in general_ first. Done correctly, 1/oo = 0
would then
> follow
> notice my disclaimer before I state that the result could
be 0.
> If you were given a test
> 1/oo =
> a/ 1
> b/ 0
> c/ oo
> d/ 10
> The answer is 0. Why? because the questioner put the
question into a
> form that clarifies division by 0 is allowable syntax.
> Given the denominator can accept oo as a construct, the
answer is 0.
> I assume this is the logic behind Rheimann sphere, its just
locating
the
> electron due north in the electron cloud isnt it?
> No.
if you can divide by oo, then the answer is 0.
T/F ?
Herc
===
Subject: Re: VOTE on whether 1/oo = 0
> I do have a liberal perspective on the topic also.
> IF you define division by infinity to start 
with
> THEN you could assign that result as 0.
> I advise against doing that.
> Its better to define the elements of the 
extension and the
> appropriate operations _in general_ first. Done correctly,
1/oo = 0
> would then follow
> notice my disclaimer before I state that the result could
be 0.
> If you were given a test
> 1/oo =
> a/ 1
> b/ 0
> c/ oo
> d/ 10
> The answer is 0. Why? because the questioner put the
question into
> a form that clarifies division by 0 is allowable syntax.
> Given the denominator can accept oo as a construct, the
answer is 0.
> I assume this is the logic behind Rheimann sphere, its just
locating
> the electron due north in the electron cloud isnt it?
> No.
> if you can divide by oo, then the answer is 0.
> T/F ?
T
which has nothing to do with locating the electron due north
in the
electron cloud.
DWC
===
Subject: Re: VOTE on whether 1/oo = 0
> I do have a liberal perspective on the topic also.
> IF you define division by infinity to start 
with
> THEN you could assign that result as 0.
> I advise against doing that.
> Its better to define the elements of the 
extension and the
> appropriate operations _in general_ first. Done correctly,
1/oo =
0
> would then follow
> notice my disclaimer before I state that the result could
be 0.
> If you were given a test
> 1/oo =
> a/ 1
> b/ 0
> c/ oo
> d/ 10
> The answer is 0. Why? because the questioner put the
question
into
> a form that clarifies division by 0 is allowable syntax.
> Given the denominator can accept oo as a construct, the
answer is
0.
> I assume this is the logic behind Rheimann sphere, its just
locating
> the electron due north in the electron cloud isnt it?
> No.
> if you can divide by oo, then the answer is 0.
> T/F ?
> which has nothing to do with locating the electron due
north in the
> electron cloud.
Results 1 - 8 of about 9 for riemann sphere electron spin.
(0.33
seconds)
Herc
===
Subject: Re: VOTE on whether 1/oo = 0
> I do have a liberal perspective on the topic also.
> IF you define division by infinity to start 
with
> THEN you could assign that result as 0.
> I advise against doing that.
> Its better to define the elements of the 
extension and the
> appropriate operations _in general_ first. Done correctly,
1/oo
= 0
> would then follow
> notice my disclaimer before I state that the result could
be 0.
> If you were given a test
> 1/oo =
1/oo = 0.125
8 is 8 at any angle, even repose.
However, limit 1/N is 0 as N grows without bound.
However, note:
Theorem: The limit of sin x/n as n grows without bound is 6.
George W. Cherry
===
Subject: Re: question of i.
> Hmm, out of curiousity--how is i defined, then?
> I once thought that it was defined by, i is a number such
that i^2
=
> -1. But, I see that once you have that defined, the same
holds true
for
> -i....so... how do you define i?
> Theres a sense in which you cant - a sense 
in which every
statement
> in the language of real numbers thats true for i is also
true for
-i.
> You could define it to be the guy one unit above the origin.
> --
> Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
But then, to define it that way... dont you 
need to have a
complex
plane defined?... for which you need i?
I thought about it for a while... it seems to me that
distinction
between i and -i is more or less arbitrary. Is it true in
general, that
wherever we see i, if we substitute -i, then everything stays
the same?
(Kind of like the difference between left-handed coordinate
system and
right-handed coordinate system.) I have a few examples where
this seems
to hold.... but anyone have a counter-example?
i) i raised to some integer power behaves the same way -i
raised to the
same integer power: e.g. i^3 = -i, (-i)^3 = i. So, it changes
to its
counter-part when raised to 3rd power (or raised to any
integer n, for
which, n % 4 = 3). And i^4 = 1, the same way (-i)^4 = 1.
ii) For roots of polynomial, theres a theorem (forgot the
name) that
says, whenever x+iy is a root, x-iy is also a root.
===
Subject: Re: question of i.
> Hmm, out of curiousity--how is i defined, then?
> I once thought that it was defined by, i is a number such
that i^2
> -1. But, I see that once you have that defined, the same
holds true
> for
> -i....so... how do you define i?
> Theres a sense in which you cant - a sense 
in which every
statement
> in the language of real numbers thats true for i is also
true for
> -i.
> You could define it to be the guy one unit above the origin.
> --
> Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
> But then, to define it that way... dont you 
need to have a
complex
> plane defined?... for which you need i?
> I thought about it for a while... it seems to me that
distinction
> between i and -i is more or less arbitrary. Is it true in
general, that
> wherever we see i, if we substitute -i, then everything
stays the same?
> (Kind of like the difference between left-handed coordinate
system and
> right-handed coordinate system.) I have a few examples
where this seems
> to hold.... but anyone have a counter-example?
> i) i raised to some integer power behaves the same way -i
raised to the
> same integer power: e.g. i^3 = -i, (-i)^3 = i. So, it
changes to its
> counter-part when raised to 3rd power (or raised to any
integer n, for
> which, n % 4 = 3). And i^4 = 1, the same way (-i)^4 = 1.
> ii) For roots of polynomial, theres a theorem (forgot the
name) that
> says, whenever x+iy is a root, x-iy is also a root.
Thats pretty much what I said:
Theres a sense in which every statement in the language of
real
numbers thats true for i is also true for -i.
Your theorem has a hypothesis, of course - that all the
coefficients
of the polynomial be real.
You can go this way. Let C be R^2 (the set of ordered pairs of
real numbers) with the usual addition (that is, if z = (a, b)
and w = (c, d) then z + w = (a+c, b+d)) and with
multiplication
defined as follows: if z = (a, b) and w = (c, d) then
zw = (ac - bd, ad + bc). Well, Ive just defined 
the complex
plane, and I havent mentioned i. I can now 
define i to be
(0, 1). Of course, I could equally well define i to be (0, 
-1),
but then all my friends would look at me funny.
Or you can write R[x] for the ring of polynomials with real
coefficients, let I be the ideal generated by the polynomial
x^2 + 1, let C be the quotient R[x]/I, and then define i to be
x + I, the coset of I containing x. You can instead define i
to be -x + I, the coset of I containing -x, but people will
laugh at you.
--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
===
Subject: Re: tell me if this is right.
This is probably not a good advice -_-;, but just for
checking answers,
most scientific calculators will be able to display fractions
(given
that the denominator is neither too big nor too small....).
Also, one
way to check if your answer is correct is to find a decimal
representation of your final answer (or... two 7 sig. 
fig.
should be
enough), and find the decimal representaion of the sum of all
the
fractions you have (you could do it by finding the dec. rep.
of each
fraction first and then additing them... or whatever suits
you)--if
your answer matches to 5 sig. fig. or so, then 
its probably
right (the
last few sig. fig.s can be ignored since there 
is such thing
as
round-off error).
> First off I want to thank everyone for your help. Math has
always
been
> my weakest subject and it bothers me that I am so bad at
it. I want
to
> change that, Im sick of fearing that I 
wouldnt pass
college algebra
> and having it hold me back from studying a wide range of
other
non-math
> things.
> Anyway, I picked up Bob Millers Algebra For The Clueless,
and have
> just been working through the the review section in the
back. One
area
> that he says should be well practiced is adding large
denominators as
> that proccess is used a lot in Algebra.
> So I practice, please tell me if I am correct.
> 7/18
> 6/24
> 3/57
> + 8/44
> -------
> = 6571/7524
> The work.
> I) Factor out all the primes.
> 7/18 = 2*3*3
> 6/24 = 2*2*2*3
> 3/57 = 3*19
> 8/44 = 2*2*11
> II) Find the LCD.
> The LCD is the product of the most number of times a prime
> appears in any one denominator So, we get the following LCD.
> LCD = 2*2*2*3*3*11*19
> Because we have 2, 3, 11, 19, as our primes
> 2 appears in 6/24 2*2*2
> 3 appears in 7/18 3*3
> 11 appears in 8/44 1
> 19 appears in 3/57 1
> LCD=2*2*2*3*3*11*19
> Am I right so far?
> (III) Multiply the top and bottom by Ôwhats 
missing.
> 7/18 2*2*2*3*3*11*19
> 2*3*3
> 2*2*19*11 <----that is the Ôwhats missing 
part.
> 7 * 2*2*11*19 5852
> ------------------- = ---------
> 2*3*3 * 2*2*11*19 15,048
> (IV) Do the same for each resulting in the following:
> 7/18 = 5852/15,048
> 6/24 = 3762/15,048
> 3/57 = 792/15,048
> 8/44 = 2736/15,048
> -------------------
> (V) Add and and reduce.
> = 13,142/15,048
> Now to reduce ive applied the Euclidean algorithm
> and got the following.
> 13142 - 0 * 15048 = 13142
> 15048 - 1 * 13142 = 1906
> 13142 - 6 * 1906 = 1706
> 1906 - 1 * 1706 = 200
> 1706 - 8 * 200 = 106
> 200 - 1 * 106 = 94
> 106 - 1 * 94 = 12
> 94 - 7 * 12 = 10
> 12 - 1 * 10 = 2
> 10 - 5 * 2 = 0
> The GCD of the two numbers is 2.
> Now since 2 will only divide into 13142 twice the final
answer when
> applied to both:
> 6571/7524
> Rob
===
Subject: Re: The myth of the beginning of time
> Time
> Van
Dont inquiring minds also want to know what the END of Time
is?
===
Subject: Re: on quaternions and octonions
>
> 4) Do Conway and Sloane give an interesting counterexample
> to unique prime factorization for Lipschitz integral
> quaternions? If so, what is it?
>> Ive found none.
> Thats too bad. I figure they must not satisfy 
a theorem
> like the above one for Hurwitz integral quaternions, or
> Conway and Smith would mention it. Also, one can see that
> the usual Euclidean algorithm fails for Lipschitz integral
> quaternions.
> So...
> Id be very happy if anyone could show that factorization
> on a given model is not unique up to unit-migration for the
> Lipschitz integral quaternions, where the quoted phrase is
> defined as above, and the Lipschitz integral quaternions are
> simply those of the form
> a + b + cj + dk
> where a,b,c,d are integers.
> Ill cite the 5 first people who come up with
counterexamples!
Lipschitz pointed out examples himself according to 
Fensters
[..] To factor, A = -1+3i+1j+2k, for example, he found triples
satisfying x^2 + y^2 + z^2 = 0 (mod 3) and x^2 + y^2 + z^2 =
0 (mod 5),
determined the associated integral quaternions with norms
three and five,
and obtained the two factorizations
-1 + 3i + 1j + 2k = (1 + 2i)(1 + i + j) = -(1 - i + j)(1 -
2k).
The above equation not only illustrates LIPSCHITZs comment
regarding
the importance of the order of the factors but also shows the
absence
of unique factorization in his proposed system of integral
quaternions. Switching the order of the factors yields a
different
quaternion. For example, -(1-2k)(1-i+j) = -1-i-3j+2k !=
-1+3i+1j+2k =
= -(1-i+j)(1-2k). The lack of unique factorization seems to
grow
worse when four divides the norm of the integral quaternion in
question. In this case, with twenty-four renditions of an
integral
prime quaternion of norm 4, there are equally many
factorizations.
To his credit, however, this proposed system represented the
first
attempt at establishing an arithmetic for the quaternions.
Moreover,
as demonstrated in the remarks cited above, he studied the
two most
analogous examples available at the time -- the arithmetics
of the
complex and the algebraic numbers -- and drew pieces of his
strategy
from each of them. When ADOLF HURWITZ considered the problem
of the
arithmetic of the quaternions a decade later, he could remark
that
``Mr. LIPSCHITZ has already developed an arithmetic of
quaternions
[HURWITZ, 1896b, 313].
-Bill Dubuque
[1] Fenster, Della Dumbaugh (1-RICH-CM)
Leonard Eugene Dickson and his work in the arithmetics of
algebras.
Arch. Hist. Exact Sci. 52 (1998), no. 2, 119-159.
http://springerlink.metapress.com/link.asp?id=fhppff3l71w9qaee
===
Subject: Re: Now I ask you:
> In free fall, who was the first scientist to discover that
the ratio
> of the distance fallen [s], divided by the square of the
elapsed time
> [t]; so that s/t^2 equals about 16 feet divided by the time
[t]
> squared; is a constant?
> Was it Galileo, or Newton?
Why do you ask if you wont listen?
===
Subject: Re: Now I ask you:
What he cant grasp is the 1/2 b/c it has no units, its a
constant
that comes from integration. I wonder what he would calculate
for the
velocity of something 2 seconds after it is dropped from
rest? By his
reasoning it would be 32 ft/s.
===
Subject: Sin Cos Tan, why not Sin Sec Tan?
I am learning trigonometry in preperation for actually
teaching it.
I am enjoying it, and would like to think I am getting a good
understanding, however I am unsure about hte following.
When talking about highschool level trigonometry we often use
ÔSOHCAHTOA as a way to remember that:
Sin@ = O/H
Cos@ = A/H
Tan@ = O/A
Further on we learn that 3 other functions exist that are the
inversion
of the first three
CSC@ = H/O
SEC@ = H/A
COT@ = A/O
So that Sin@ = 1/CSC@
Cos@ = 1/SEC@
Tan@ = 1/COT@
My question is why is the cofunction of Sin, ie Cosine placed
in the
first three that are learnt. Wouldnt it make 
more sense to
group them as
Sin@ = O/H
Sec@ = H/A
Tan@ = O/A
Then introduce the cofunctions
Cos@ = A/H
CSC@ = H/O
Cot@ = A/O
This seems alot more clear to me. Is there some mathematical
reason that
I am missing?
Cassandra Thompson
===
Subject: Re: Sin Cos Tan, why not Sin Sec Tan?
|I am learning trigonometry in preperation for actually
teaching it.
|I am enjoying it, and would like to think I am getting a good
|understanding, however I am unsure about hte following.
|
|When talking about highschool level trigonometry we often use
|SOHCAHTOA as a way to remember that:
|Sin@ = O/H
|Cos@ = A/H
|Tan@ = O/A
|
|Further on we learn that 3 other functions exist that are
the inversion
|of the first three
|
|CSC@ = H/O
|SEC@ = H/A
|COT@ = A/O
|
|So that Sin@ = 1/CSC@
| Cos@ = 1/SEC@
| Tan@ = 1/COT@
|
|
|My question is why is the cofunction of Sin, ie Cosine
placed in the
|first three that are learnt. Wouldnt it make 
more sense to
group them as
|
|Sin@ = O/H
|Sec@ = H/A
|Tan@ = O/A
|
|Then introduce the cofunctions
|Cos@ = A/H
|CSC@ = H/O
|Cot@ = A/O
|
|This seems alot more clear to me. Is there some mathematical
reason that
|I am missing?
the name trigonometry suggests that the subject is all about
triangles, but thats very misleading; what trigonometry
secretly
_really_ is is the study of the points on the unit circle.
(the right
triangles that show up are just auxiliary devices used to
highlight
the points on the unit circle.) from this point of view its
pretty
clear why cosine and sine are the crucial variables; theyre
the x and
y coordinates of the point on the unit circle.
so in fact besides not bothering with secant and cosecant and
cotangent, its probably better not to bother with tangent
either.
and cosine should generally come before sine, of course,
since x comes
before y after all.
of course this tends to reduce trigonometry to about a
five-minute
lesson, but thats probably about what its 
worth.
--
[e-mail address jdolan@math.ucr.edu]
===
Subject: Re: Sin Cos Tan, why not Sin Sec Tan?
There are two different gouping schemes going on here:
The reciprocal group sin/csc, cos/sec, and tan/cot
The cofunction group sin/cos, sec/csc, and tan/cot
Parenthetical Warning: Note tan and cot are both reciprocals
and
cofunctions and are the only pair with this property.
Students will often
jump to conclusions about the other pairs. Something to watch
out for.
We can choose from either group for our emphasis.
Traditionally
sin/cos/tan
are chosen. I suppose sin/sec/tan would work but I feel there
are
advantages to the first.
All of sin/sec/tan are increasing functions for first quadrant
angles. It
is often convenient to have a decreasing function to model
behaviors, thus
cos is likely to be of advantage.
The identity that I told my students they would remember to
tell their
grandchildren, sin^2 +cos^2 = 1 would be clunky as
sin^2 + 1/sec^2 = 1 and would have the disadvantage of not
being valid at
multiples of pi/2.
By the way, one way to interpret the co-functions is as
follows: cos(
angle) is the COmplements Sine. Similarly for the other
pairs of
co-functions. I am not sure if that is just a memory device
or is lurking
in the actual naming of the functions originally.
Having the students adept with all the trig functions is
essential. They
have to think of these as their friends.
Some replies have suggested that sec and csc are hardly ever
used later. I
have to disagree. In calc, integrals involving the
sqrt(u^2-1) yield to the substitution u = sec(theta) for
their evaluation.
Other approaches are also possible.
Just my wandering 2 cents worth.
Ken
> I am learning trigonometry in preperation for actually
teaching it.
> I am enjoying it, and would like to think I am getting a
good
> understanding, however I am unsure about hte following.
> When talking about highschool level trigonometry we often
use
> ÔSOHCAHTOA as a way to remember that:
> Sin@ = O/H
> Cos@ = A/H
> Tan@ = O/A
> Further on we learn that 3 other functions exist that are
the inversion
> of the first three
> CSC@ = H/O
> SEC@ = H/A
> COT@ = A/O
> So that Sin@ = 1/CSC@
> Cos@ = 1/SEC@
> Tan@ = 1/COT@
> My question is why is the cofunction of Sin, ie Cosine
placed in the
> first three that are learnt. Wouldnt it make 
more sense to
group them as
> Sin@ = O/H
> Sec@ = H/A
> Tan@ = O/A
> Then introduce the cofunctions
> Cos@ = A/H
> CSC@ = H/O
> Cot@ = A/O
> This seems alot more clear to me. Is there some
mathematical reason that
> I am missing?
> Cassandra Thompson
===
Subject: Re: Sin Cos Tan, why not Sin Sec Tan?
When sin,cos are taken up first, the idea is to acquaint them
to the
student as ratios of sides. The denomintor is the largest side
hypotenuse, so sin, cos values are <1, is bounded. Tan
function is sin
divided by cos and can take any value between + and -
infinity. This
matter is slowly introduced to the student.
However, I also felt a sort of discontinuity when a I first
learnt
Trig in school, same what you are referring to.
After the usual sin,cos,tan we introduce the other three
functions,
as reciprocal functions AND ALSO as CO- functions. There is a
small
and temporary learning block there.
I suggest after after sin,cos,tan introduce them using
acronyms
Counter Clockwise
[ acronym say, Secondary School Certificate ]
Sin@ = O/H
Sec@ = H/A
Cot@ = A/O
and, Clockwise
[ acronym say, Texas Cricket Club, may not exist :).. ]
Tan@ = O/A
Cos@ = A/H
Csc@ = H/O
with that, all the 9 quantities would have been learnt up
with their
inter-relationship.
> I am learning trigonometry in preperation for actually
teaching it.
> I am enjoying it, and would like to think I am getting a
good
> understanding, however I am unsure about hte following.
> When talking about highschool level trigonometry we often
use
> ÔSOHCAHTOA as a way to remember that:
> Sin@ = O/H
> Cos@ = A/H
> Tan@ = O/A
> Further on we learn that 3 other functions exist that are
the inversion
> of the first three
> CSC@ = H/O
> SEC@ = H/A
> COT@ = A/O
> So that Sin@ = 1/CSC@
> Cos@ = 1/SEC@
> Tan@ = 1/COT@
> My question is why is the cofunction of Sin, ie Cosine
placed in the
> first three that are learnt. Wouldnt it make 
more sense to
group them as
> Sin@ = O/H
> Sec@ = H/A
> Tan@ = O/A
> Then introduce the cofunctions
> Cos@ = A/H
> CSC@ = H/O
> Cot@ = A/O
> This seems alot more clear to me. Is there some
mathematical reason that
> I am missing?
> Cassandra Thompson
===
Subject: Re: Sin Cos Tan, why not Sin Sec Tan?
> I am learning trigonometry in preperation for actually
teaching it.
> I am enjoying it, and would like to think I am getting a
good
> understanding, however I am unsure about hte following.
> When talking about highschool level trigonometry we often
use
> ÔSOHCAHTOA as a way to remember that:
> Sin@ = O/H
> Cos@ = A/H
> Tan@ = O/A
> Further on we learn that 3 other functions exist that are
the inversion
> of the first three
> CSC@ = H/O
> SEC@ = H/A
> COT@ = A/O
> So that Sin@ = 1/CSC@
> Cos@ = 1/SEC@
> Tan@ = 1/COT@
> My question is why is the cofunction of Sin, ie Cosine
placed in the
> first three that are learnt. Wouldnt it make 
more sense to
group them as
> Sin@ = O/H
> Sec@ = H/A
> Tan@ = O/A
> Then introduce the cofunctions
> Cos@ = A/H
> CSC@ = H/O
> Cot@ = A/O
> This seems alot more clear to me. Is there some
mathematical reason that
> I am missing?
Tradition mainly; also cosine does come up naturally more
often that
sec/cosec/cot, notably the cosine rule in trigonometry. I
dislike
the mnemonics for these --- one has to learn the mnemonic as
well
as the facts, but it helps to know that the co-thing is the
thing of
the co-mplementary angle, e.g., cosecant of x is the secant of
the complement of x. Also the non-co functions, sin, sec and
tan
are those that are increasing on acuate angles (and the cos
are decreasing).
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square
root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: Re: Sin Cos Tan, why not Sin Sec Tan?
> Sin@ = O/H
> Cos@ = A/H
These 2 are grouped together because they divide by the
longest
side on the triangle and are both <=1.
But Tan is useful for the simplest of geometrical calculations
so it became a 3some.
> Tan@ = O/A
Either Tan or Sin&Cos can be taught 1st, IMO.
Herc
===
Subject: Re: Sin Cos Tan, why not Sin Sec Tan?
>>Sin@ = O/H
>>Cos@ = A/H
> These 2 are grouped together because they divide by the
longest
> side on the triangle and are both <=1.
> But Tan is useful for the simplest of geometrical
calculations
> so it became a 3some.
>>Tan@ = O/A
> Either Tan or Sin&Cos can be taught 1st, IMO.
> Herc
NB.
I should add that you have cleared it up somewhat.
You are in effect saying (regardless of my slight confusion
as to why
sin/cos is better to teach first then sec/csc) that we are
teaching the
students:
(sin/cos) and (tan)
as opposed to
(sin, cos, tan)
Is this correct?
===
Subject: Re: Sin Cos Tan, why not Sin Sec Tan?
>
>>Sin@ = O/H
>>Cos@ = A/H
>
>
> These 2 are grouped together because they divide by the
longest
> side on the triangle and are both <=1.
>
> But Tan is useful for the simplest of geometrical
calculations
> so it became a 3some.
>
>
>>Tan@ = O/A
>
>
> Either Tan or Sin&Cos can be taught 1st, IMO.
>
> Herc
>
>
>
> NB.
> I should add that you have cleared it up somewhat.
> You are in effect saying (regardless of my slight confusion
as to why
> sin/cos is better to teach first then sec/csc) that we are
teaching the
> students:
> (sin/cos) and (tan)
> as opposed to
> (sin, cos, tan)
> Is this correct?
The main aim of teaching is to captivate the interest of the
kids for
long enough in some imaginative way to them to the point
where you can
tell/ask them to go figure.
Re Trig (sin/cos)  build a small model consisting of a clear
round
disk with a 100mm radius. Stick a pin through the centre of
the disk
and a pin into the edge of said disk. Now place the disk on
some mm
graph paper  so the disk can rotate around the centre pin. Tie
a
cotton thread to the centre pin and loop the thread around
the pin on
the edge  hold the untied end in such a manner that it is
always at
right angles to the x-axis.
Now rotate the disk form 0 through 90 degrees while still
holding the
untied end of the thread at right angles to the x-axis  as the
angle
increases the distance of the thread on the x-axis (adjacent)
to the
centre point decreases and the edge pin height increase with
respect
to the y-axis (opposite).
The interesting bit to note is that when the disk has been
rotated for
say 30 degrees the x-axis (adjacent) will read 0.866 and the
y-axis
(opposite) 0.5. Playing with this for a bit should get a
couple of
good points across (Sine waves etc)  adding the x and y axis
numbers
together is also interesting  they should find that the
highest
number achievable is 1.414 and at an angle of 45 degree.
Now the go figure bit (apart from the boring 
obvious
like knowing
two of the components the third can be calculated) for
instance
- at
what angle will a cannon shoot a projectile the greatest
distance? Why
doesnt NASA position rockets at this angle and save a bunch
of fuel?
And so on etc.
Bill
===
Subject: Re: Sin Cos Tan, why not Sin Sec Tan?
>
>>Sin@ = O/H
>>Cos@ = A/H
>
>
> These 2 are grouped together because they divide by the
longest
> side on the triangle and are both <=1.
>
> But Tan is useful for the simplest of geometrical
calculations
> so it became a 3some.
>
>
>>Tan@ = O/A
>
>
> Either Tan or Sin&Cos can be taught 1st, IMO.
>
> Herc
>
>
>
> NB.
> I should add that you have cleared it up somewhat.
> You are in effect saying (regardless of my slight confusion
as to why
> sin/cos is better to teach first then sec/csc) that we are
teaching the
> students:
> (sin/cos) and (tan)
> as opposed to
> (sin, cos, tan)
> Is this correct?
The main aim of teaching is to captivate the interest of the
kids for
long enough in some imaginative way to the point where you can
tell/ask them to go figure.
Re Trig (as an example)  build a small model consisting of a
clear
round disk with a 100mm radius. Stick a pin through the
centre of the
disk and a pin into the edge of said disk. Now place the disk
on some
mm graph paper  so the disk can rotate around the centre pin.
Tie a
cotton thread to the centre pin and loop the thread around
the pin on
the edge  hold the untied end in such a manner that it is
always at
right angles to the x-axis.
Now rotate the disk form 0 through 90 degrees while still
holding the
untied end of the thread at right angles to the x-axis  as the
angle
increases the distance of the thread on the x-axis (adjacent)
to the
centre point decreases and the edge pin height increase with
respect
to the y-axis (opposite).
The interesting bit to note is that when the disk has been
rotated for
say 30 degrees the x-axis (adjacent) will read 0.866 and the
y-axis
(opposite) 0.5. Playing with this for a bit should get a
couple of
good points across  adding the x and y axis numbers together
is
also
interesting  they should find that the highest number
achievable is
1.414 and at an angle of 45 degree.
Now the go figure bit (apart from the boring 
obvious
like knowing
two of the components the third can be calculated) for
instance
- at
what angle will a cannon shoot a projectile the greatest
distance? Why
doesnt NASA position rockets at this angle and save a bunch
of fuel?
And so on etc.
Bill
===
Subject: Re: Sin Cos Tan, why not Sin Sec Tan?
>>Sin@ = O/H
>>Cos@ = A/H
> These 2 are grouped together because they divide by the
longest
> side on the triangle and are both <=1.
> But Tan is useful for the simplest of geometrical
calculations
> so it became a 3some.
>>Tan@ = O/A
> Either Tan or Sin&Cos can be taught 1st, IMO.
> Herc
> NB.
> I should add that you have cleared it up somewhat.
> You are in effect saying (regardless of my slight confusion
as to why
> sin/cos is better to teach first then sec/csc) that we are
teaching the
> students:
> (sin/cos) and (tan)
> as opposed to
> (sin, cos, tan)
> Is this correct?
Yes.
This is probably the main underpinning of the preference.
Results 1 - 10 of about 1,820 for sin wave.
but there are many others some mentioned already. In virtually
any engineering or computing field you will use sin routinely.
Also d/dx sin(x) = cos(x)
We did months on just Tan in year 10, and in year 11 we did a
unit on
circles
with mostly just sin and cos and different definitions that
sin@ = y and
cos@ = x.
(on the unit circle).
Sin describes vibrations, propogation, directions, .. it
applies to the real
world.
To use the inverse trig functions youre probably working
things out
backwards.
Like www.searchsky.com it plots the stars locations onto the
screen.
Inverse functions for an inverse application!! I just checked
the code
has about
100 sin and 10 asin. This is the section to approximate the
moon position.
I mean reciprocal not inverse.. but anyway same argument.
L1 = trange(280.466 + 36000.8 * t);
M1 = trange(357.529+35999*t - 0.0001536* t*t +
t*t*t/24490000);
C1 = (1.915 - 0.004817* t - 0.000014* t * t)* dsin(M1);
C1 = C1 + (0.01999 - 0.000101 * t)* dsin(2*M1);
C1 = C1 + 0.00029 * dsin(3*M1);
V1 = M1 + C1;
Ec1 = 0.01671 - 0.00004204 * t - 0.0000001236 * t*t;
R1 = 0.99972 / (1 + Ec1 * dcos(V1));
Th1 = L1 + C1;
Om1 = trange(125.04 - 1934.1 * t);
Lam1 = Th1 - 0.00569 - 0.00478 * dsin(Om1);
Obl = (84381.448 - 46.815 * t)/3600;
Ra1 = datan2(dsin(Th1) * dcos(Obl) - dtan(0)* dsin(Obl),
dcos(Th1));
Dec1 = dasin(dsin(0)* dcos(Obl) +
dcos(0)*dsin(Obl)*dsin(Th1));
F = trange(93.2721 + 483202 * t - 0.003403 * t* t - t * t *
t/3526000);
L2 = trange(218.316 + 481268 * t);
Om2 = trange(125.045 - 1934.14 * t + 0.002071 * t * t + t * t
* t/450000);
M2 = trange(134.963 + 477199 * t + 0.008997 * t * t + t * t *
t/69700);
D = trange(297.85 + 445267 * t - 0.00163 * t * t + t * t *
t/545900);
D2 = 2*D;
R2 = 1 + (-20954 * dcos(M2) - 3699 * dcos(D2 - M2) - 2956 *
dcos(D2)) /
385000;
R3 = (R2 / R1) / 379.168831168831;
Bm = 5.128 * dsin(F) + 0.2806 * dsin(M2 + F);
Bm = Bm + 0.2777 * dsin(M2 - F) + 0.1732 * dsin(D2 - F);
Lm = 6.289 * dsin(M2) + 1.274 * dsin(D2 -M2) + 0.6583 *
dsin(D2);
Lm = Lm + 0.2136 * dsin(2*M2) - 0.1851 * dsin(M1) - 0.1143 *
dsin(2 * F);
Lm = Lm +0.0588 * dsin(D2 - 2*M2)
Lm = Lm + 0.0572* dsin(D2 - M1 - M2) + 0.0533* dsin(D2 + M2);
Lm = Lm + L2;
Ra2 = datan2(dsin(Lm) * dcos(Obl) - dtan(Bm)* dsin(Obl),
dcos(Lm));
Dec2 = dasin(dsin(Bm)* dcos(Obl) +
dcos(Bm)*dsin(Obl)*dsin(Lm));
HLm = trange(Lam1 + 180 + (180/Math.PI) * R3 * dcos(Bm) *
dsin(Lam1 -
Lm));
HBm = R3 * Bm;
I = 1.54242;
W = Lm - Om2;
Y = dcos(W) * dcos(Bm);
X = dsin(W) * dcos(Bm) * dcos(I) - dsin(Bm) * dsin(I);
A = datan2(X, Y);
EL = A - F;
EB = dasin(-dsin(W) * dcos(Bm) * dsin(I) - dsin(Bm) *
dcos(I));
W = trange(HLm - Om2);
Y = dcos(W) * dcos(HBm);
X = dsin(W) * dcos(HBm) * dcos(I) - dsin(HBm) * dsin(I);
A = datan2(X, Y);
SL = trange(A - F);
SB = dasin(-dsin(W) * dcos(HBm) * dsin(I) - dsin(HBm) *
dcos(I));
not too complicated to be accurate to 10,000 years!
Herc
===
Subject: Re: Sin Cos Tan, why not Sin Sec Tan?
Cassandra:
Some decisions in math are arbitrary. For example, you
occassionally get
people posting a statement that pi as 3.14159... shouldnt 
be
the
fundamental constant; it should be half of this (1.57...) or
double this
(6.283...), as these would make some expressions slightly
simpler.
Maybe the 6 legged octopuses that live on planet Xoen use
1.57.. or 6.283..
as the fundamental constant, and on Xoen pi is defined as the
ratio of a
circles circumference to its radius. I dont 
know.
However, I bet on Xoen they use sin and cos and not sec and
csc:
1. You far more often see sin in a numerator than a
denominator, so sin
should be the basic ratio and not 1/sin.
2. This is partly because of a point somebody else made, that
when you
decompose a vector into its components, sin and cos appear
naturally. This
is a lot of the practical (and theoretical) use of trig.
3. Pythagorases thereom is sin^2 + cos^2 = 1. This is used
all the time.
Its
a lot easier to spot and use than 1/sec^2 + 1/csc^2 = 1.
You certainly could use csc and sec instead of sin and cos.
But consider
the
use that you make of trig, and look at what the corresponding
formulas
would
be if you used sec and csc. I bet 90%+ the formulas are
simpler using sin
and cos, and when you start doing calculus, this ratio
increases to 95%+.
>>Sin@ = O/H
>>Cos@ = A/H
> These 2 are grouped together because they divide by the
longest
> side on the triangle and are both <=1.
> But Tan is useful for the simplest of geometrical
calculations
> so it became a 3some.
>>Tan@ = O/A
> Either Tan or Sin&Cos can be taught 1st, IMO.
> Herc
> NB.
> I should add that you have cleared it up somewhat.
> You are in effect saying (regardless of my slight confusion
as to why
> sin/cos is better to teach first then sec/csc) that we are
teaching the
> students:
> (sin/cos) and (tan)
> as opposed to
> (sin, cos, tan)
> Is this correct?
===
Subject: Re: Sin Cos Tan, why not Sin Sec Tan?
>>Sin@ = O/H
>>Cos@ = A/H
> These 2 are grouped together because they divide by the
longest
> side on the triangle and are both <=1.
> But Tan is useful for the simplest of geometrical
calculations
> so it became a 3some.
>>Tan@ = O/A
> Either Tan or Sin&Cos can be taught 1st, IMO.
> Herc
You have touched on what I was trying to say in a previous
post (one
that was about the same time as yours).
Why is it important that the hypotonuse be the denomitator?
Because the
answer will always be <=1. Because as far as I can tell that
is the only
reason for choosing the sin/cos pair over the sec/csc pair.
I am guessing that there is a very good answer why this is
better then
the result being >1, however I cannot yet see why.
Cassandra Thompson
===
Subject: Re: Sin Cos Tan, why not Sin Sec Tan?
...
> You have touched on what I was trying to say in a previous
post (one
> that was about the same time as yours).
> Why is it important that the hypotonuse be the denomitator?
Because the
> answer will always be <=1. Because as far as I can tell
that is the only
> reason for choosing the sin/cos pair over the sec/csc pair.
> I am guessing that there is a very good answer why this is
better then
> the result being >1, however I cannot yet see why.
Because in that case you have angles for which the function
value is not
defined.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam,
nederland,
+31205924131
home: bovenover 215, 1025 jn amsterdam, nederland;
http://www.cwi.nl/~dik/
===
Subject: Re: Sin Cos Tan, why not Sin Sec Tan?
Originator: dwildstr@zeno.ucsd.edu (Jake Wildstrom)
The Prophet Cassandra Thompson known to the wise as
cass.harley@bigpond.com, opened the Book of Words, and read
unto the people:
>Why is it important that the hypotonuse be the denomitator?
Because the
>answer will always be <=1. Because as far as I can tell that
is the only
>reason for choosing the sin/cos pair over the sec/csc pair.
>I am guessing that there is a very good answer why this is
better then
>the result being >1, however I cannot yet see why.
Well, bounded functions are frequently nicer. In calculus and
analysis, youll find there are many things 
which sin and cos
naturally fall out of (for instance, theres a grand cosmic
relationship between the exponential function e^x and the sin
and cos
functions; also, the sin and cos functions, unlike the sec
and csc
functions, have good approximating sequences and behave
generally
nicely in ways which sec and csc fail to do. One last,
simpler,
example: sin and cos are defined everywhere; sec and csc have
discontinuities when they jump to infinity, which is really
unpleasant
(tan has this problem too, but with the tan/cot pair, this is
unavoidable; the distinction of one of tan/cot as 
Ôcustomary
is
moderately arbitrary, but the distinction of sin/cos as
customary and
sec/csc as exotic is in fact a nonarbitrary decision).
+------------------------------------------------------------
-+
| D. Jacob Wildstrom -- Math monkey and freelance thinker |
| Graduate Student, University of California at San Diego |
| A mathematician is a device for turning coffee into |
| theorems. -Alfred Renyi |
+------------------------------------------------------------
-+
The opinions expressed herein are not necessarily endorsed by
the
University of California or math department thereof.
===
Subject: Re: Sin Cos Tan, why not Sin Sec Tan?
> The Prophet Cassandra Thompson known to the wise as
cass.harley@bigpond.com, opened the Book of Words, and read
unto the people:
>>Why is it important that the hypotonuse be the denomitator?
Because the
>>answer will always be <=1. Because as far as I can tell
that is the only
>>reason for choosing the sin/cos pair over the sec/csc pair.
>>I am guessing that there is a very good answer why this is
better then
>>the result being >1, however I cannot yet see why.
> Well, bounded functions are frequently nicer. In calculus
and
> analysis, youll find there are many things 
which sin and cos
> naturally fall out of (for instance, theres a grand 
cosmic
> relationship between the exponential function e^x and the
sin and cos
> functions; also, the sin and cos functions, unlike the sec
and csc
> functions, have good approximating sequences and behave
generally
> nicely in ways which sec and csc fail to do. One last,
simpler,
> example: sin and cos are defined everywhere; sec and csc 
have
> discontinuities when they jump to infinity, which is really
unpleasant
> (tan has this problem too, but with the tan/cot pair, this
is
> unavoidable; the distinction of one of tan/cot as
Ôcustomary is
> moderately arbitrary, but the distinction of sin/cos as
customary and
> sec/csc as exotic is in fact a nonarbitrary decision).
>
+------------------------------------------------------------
-+
> | D. Jacob Wildstrom -- Math monkey and freelance thinker |
> | Graduate Student, University of California at San Diego |
> | A mathematician is a device for turning coffee into |
> | theorems. -Alfred Renyi |
>
+------------------------------------------------------------
-+
> The opinions expressed herein are not necessarily endorsed
by the
> University of California or math department thereof.
clarified what I was trying to understand, and I feel like I
have a much
stronger knowledge about it.
Cassandra Thompson.
===
Subject: Re: Sin Cos Tan, why not Sin Sec Tan?
> Sin@ = O/H
> Cos@ = A/H
> Tan@ = O/A
> Further on we learn that 3 other functions exist that are
the inversion
> of the first three
> CSC@ = H/O
> SEC@ = H/A
> COT@ = A/O
> My question is why is the cofunction of Sin, ie Cosine
placed in the
> first three that are learnt.
For your students: cos and sin are often used to find the
lengths of the
non-hypoteneuse sides of a right-angled triangle, given the
angle. This
happens for example when decomposing a vector of length x at
angle @ from
the coordinate axes: one component is xsin@, and the other is
xcos@.
Theres a symmetry there that you wouldnt get 
if you used
one of the
other functions. Likewise in identities like sin^2(@) +
cos^2(@) = 1. Or
if you introduce the functions on a circle, youll show the
kids that the
sine of an arc-length in radians gives its vertical component
and the
cosine its horizontal component. Again, theres a symmetry
between the two
that it would be crazy to change. If you do lots more work
with these
functions it should become clear to you that sin and cos are
a natural
pair, whereas the relationship between sin and sec, for
example, is more
obscure. Then youll be ready to teach the kids. Good luck 
to
all
involved.
===
Subject: Re: Sin Cos Tan, why not Sin Sec Tan?
> Sin@ = O/H
> Cos@ = A/H
> Tan@ = O/A
> Further on we learn that 3 other functions exist that are
the inversion
> of the first three
> CSC@ = H/O
> SEC@ = H/A
> COT@ = A/O
> My question is why is the cofunction of Sin, ie Cosine
placed in the
> first three that are learnt.
> For your students: cos and sin are often used to find the
lengths of the
> non-hypoteneuse sides of a right-angled triangle, given the
angle. This
> happens for example when decomposing a vector of length x
at angle @ from
> the coordinate axes: one component is xsin@, and the other
is xcos@.
> Theres a symmetry there that you wouldnt 
get if you used
one of the
> other functions. Likewise in identities like sin^2(@) +
cos^2(@) = 1. Or
> if you introduce the functions on a circle, youll show 
the
kids that the
> sine of an arc-length in radians gives its vertical
component and the
> cosine its horizontal component. Again, theres a symmetry
between the
two
> that it would be crazy to change. If you do lots more work
with these
> functions it should become clear to you that sin and cos
are a natural
> pair, whereas the relationship between sin and sec, for
example, is more
> obscure. Then youll be ready to teach the kids. Good luck
to all
> involved.
Good points. Also, their derivatives involve each other:
(sin x) = cos x;
(cos x) = -sin x.
[Because the sine and cosine are just 90 degrees
(pi/2 radians) out of phase with one another.]
(So you can think of the cosine as expressing the
slope of the sine function.)
Note also that
sin(A + B) = sin A cos B + cos A sin B;
cos(A + B) = cos A cos B - sin A sin B.
So, sin x and cos x are intimately related.
WRT tan A, note that
tan A = sin A/cos A
There are just a limited number of relations
among the hypotenuse, opposite side, and
adjacent side of a right triangle, and the
cosecant, secant, and cotangent are just the
reciprocals of the sine, cosine, and tangent
respectively. Keep going back to the right
triangle and the relation between one of its
acute angles, its adjacent side, opposite
side, and hypotenuse. Stress that the length
of its hypotenuse is the square root of the
squares on the opposite two sides. Picture
that!
George
===
Subject: Re: Sin Cos Tan, why not Sin Sec Tan?
>>Sin@ = O/H
>>Cos@ = A/H
>>Tan@ = O/A
>>Further on we learn that 3 other functions exist that are
the inversion
>>of the first three
>>CSC@ = H/O
>>SEC@ = H/A
>>COT@ = A/O
>>My question is why is the cofunction of Sin, ie Cosine
placed in the
>>first three that are learnt.
> For your students: cos and sin are often used to find the
lengths of the
> non-hypoteneuse sides of a right-angled triangle, given the
angle. This
> happens for example when decomposing a vector of length x
at angle @ from
> the coordinate axes: one component is xsin@, and the other
is xcos@.
> Theres a symmetry there that you wouldnt 
get if you used
one of the
> other functions. Likewise in identities like sin^2(@) +
cos^2(@) = 1. Or
> if you introduce the functions on a circle, youll show 
the
kids that the
> sine of an arc-length in radians gives its vertical
component and the
> cosine its horizontal component. Again, theres a symmetry
between the
two
> that it would be crazy to change. If you do lots more work
with these
> functions it should become clear to you that sin and cos
are a natural
> pair, whereas the relationship between sin and sec, for
example, is more
> obscure. Then youll be ready to teach the kids. Good luck
to all
> involved.
I can see how sin and cos are a natural pair, being that
cosine is the
cofunction of sin.
I am still trying to understand why we have a tendency (as
teachers) to
present the three as if there were an inpenetrable group.
ie (sin, cos, tan)
When it appears, at least to me, that we are really trying to
teach them:
(sin and its cofunction, cos) and (tan).
(Sec, and its cofunction csc) and (cot) get a mention one the
student
fully understands the first three.
[I hope the brackets indicate how my mind groups these
functions).
Would it be just as worthwhile to the student to teach
instead.
(sec and its cofunction csc), and tan???
I know this question is more about pedagogy then mathematics,
it is
important for me to understand why we teach in the way that
we do. I
will continue working through more trigonometry, however I
imagine that
it would be in the high levels of trig that I would
eventually see why
sin and cos are more important then sec and csc. At the
moment I really
see them as being very equal, and cant understand why one
pair is
better then the other.
===
Subject: Re: Sin Cos Tan, why not Sin Sec Tan?
by the way, I still half-subconsciously use the mnemnonic I
learnt for the
sequence S O H C A H T O A (sin=opp/hyp; cos=adj/hyp;
tan=opp/adj):
Some Officers Have Curly Auburn Hair To Offer Attraction
not sure if todays kids will like that, its 
not
particularly good but
somehow it stuck in my head.
===
Subject: Re: Sin Cos Tan, why not Sin Sec Tan?
>
>
>>Sin@ = O/H
>>Cos@ = A/H
>>Tan@ = O/A
>
>
>>Further on we learn that 3 other functions exist that are
the inversion
>>of the first three
>
>
>>CSC@ = H/O
>>SEC@ = H/A
>>COT@ = A/O
>
>
>>My question is why is the cofunction of Sin, ie Cosine
placed in the
>>first three that are learnt.
>
>
> For your students: cos and sin are often used to find the
lengths of
the
> non-hypoteneuse sides of a right-angled triangle, given the
angle. This
> happens for example when decomposing a vector of length x
at angle @
from
> the coordinate axes: one component is xsin@, and the other
is xcos@.
> Theres a symmetry there that you wouldnt 
get if you used
one of the
> other functions. Likewise in identities like sin^2(@) +
cos^2(@) = 1.
Or
> if you introduce the functions on a circle, youll show 
the
kids that
the
> sine of an arc-length in radians gives its vertical
component and the
> cosine its horizontal component. Again, theres a symmetry
between the
two
> that it would be crazy to change. If you do lots more work
with these
> functions it should become clear to you that sin and cos
are a natural
> pair, whereas the relationship between sin and sec, for
example, is
more
> obscure. Then youll be ready to teach the kids. Good luck
to all
> involved.
> I can see how sin and cos are a natural pair, being that
cosine is the
> cofunction of sin.
> I am still trying to understand why we have a tendency (as
teachers) to
> present the three as if there were an inpenetrable group.
> ie (sin, cos, tan)
> When it appears, at least to me, that we are really trying
to teach them:
> (sin and its cofunction, cos) and (tan).
> (Sec, and its cofunction csc) and (cot) get a mention one
the student
> fully understands the first three.
> [I hope the brackets indicate how my mind groups these
functions).
> Would it be just as worthwhile to the student to teach
instead.
> (sec and its cofunction csc), and tan???
> I know this question is more about pedagogy then
mathematics, it is
> important for me to understand why we teach in the way that
we do. I
> will continue working through more trigonometry, however I
imagine that
> it would be in the high levels of trig that I would
eventually see why
> sin and cos are more important then sec and csc. At the
moment I really
> see them as being very equal, and cant understand why one
pair is
> better then the other.
Actually, for a right triangle with unequal shorter sides,
there are six
ratios of 2 different sides possible, and each of them has a
name. If
this is pointed out at the beginning and it is explained that
eventually
each of the six will be considered, with suitable mnemonic
help to keep
them straight, then there is a good deal less confusion among
students
newly presented with these six ratio functions.
Use complimentary angles: the compliment of an angle,
co-angle, is a
right angle minus the original angle, which for either acute
angle in a
right triangle means the other acute angle. Any function of
the co-angle
is the co-function of the original angle, which cuts it down
to 3
functions and the co-relations. Sine, cosine, secant and
cosecant
involve the hypotenuese, but the tangent and cotangent do not.
===
Subject: Re: Sin Cos Tan, why not Sin Sec Tan?
>>Sin@ = O/H
>>Cos@ = A/H
>>Tan@ = O/A
>>Further on we learn that 3 other functions exist that are
the inversion
>>of the first three
>>CSC@ = H/O
>>SEC@ = H/A
>>COT@ = A/O
>>My question is why is the cofunction of Sin, ie Cosine
placed in the
>>first three that are learnt.
>For your students: cos and sin are often used to find the
lengths of the
>non-hypoteneuse sides of a right-angled triangle, given the
angle. This
>happens for example when decomposing a vector of length x at
angle @
from
>the coordinate axes: one component is xsin@, and the other
is xcos@.
>Theres a symmetry there that you wouldnt 
get if you used
one of the
>other functions. Likewise in identities like sin^2(@) +
cos^2(@) = 1. Or
>if you introduce the functions on a circle, youll show the
kids that
the
>sine of an arc-length in radians gives its vertical
component and the
>cosine its horizontal component. Again, theres a symmetry
between the
two
>that it would be crazy to change. If you do lots more work
with these
>functions it should become clear to you that sin and cos are
a natural
>pair, whereas the relationship between sin and sec, for
example, is more
>obscure. Then youll be ready to teach the kids. Good luck
to all
>involved.
>>I can see how sin and cos are a natural pair, being that
cosine is the
>>cofunction of sin.
>>I am still trying to understand why we have a tendency (as
teachers) to
>>present the three as if there were an inpenetrable group.
>>ie (sin, cos, tan)
>>When it appears, at least to me, that we are really trying
to teach them:
>>(sin and its cofunction, cos) and (tan).
>>(Sec, and its cofunction csc) and (cot) get a mention one
the student
>>fully understands the first three.
>>[I hope the brackets indicate how my mind groups these
functions).
>>Would it be just as worthwhile to the student to teach
instead.
>>(sec and its cofunction csc), and tan???
>>I know this question is more about pedagogy then
mathematics, it is
>>important for me to understand why we teach in the way that
we do. I
>>will continue working through more trigonometry, however I
imagine that
>>it would be in the high levels of trig that I would
eventually see why
>>sin and cos are more important then sec and csc. At the
moment I really
>>see them as being very equal, and cant understand why one
pair is
>>better then the other.
> Actually, for a right triangle with unequal shorter sides,
there are six
> ratios of 2 different sides possible, and each of them has
a name. If
> this is pointed out at the beginning and it is explained
that eventually
> each of the six will be considered, with suitable mnemonic
help to keep
> them straight, then there is a good deal less confusion
among students
> newly presented with these six ratio functions.
I was just doing the dishes thing about this thread and what
you have
said is pretty much what I thought would be the way to go.
ie upfront tell them, there is sin and cos, sec and csc, tan
and cot.
The all relate to each other in various way, which we will
learn. The
ones you are most likely to use in your vocation will be
sin/cos and
tan. We will start with them....
> Use complimentary angles: the compliment of an angle,
co-angle, is a
> right angle minus the original angle, which for either
acute angle in a
> right triangle means the other acute angle. Any function of
the co-angle
> is the co-function of the original angle, which cuts it
down to 3
> functions and the co-relations. Sine, cosine, secant and
cosecant
> involve the hypotenuese, but the tangent and cotangent do
not.
===
Subject: Re: Sin Cos Tan, why not Sin Sec Tan?
> I am still trying to understand why we have a tendency (as
teachers)
> to present the three as if there were an inpenetrable group.
> ie (sin, cos, tan)
> When it appears, at least to me, that we are really trying
to teach
> them: (sin and its cofunction, cos) and (tan).
> (Sec, and its cofunction csc) and (cot) get a mention one
the student
> fully understands the first three.
If they are mentioned at all. :-) Who needs them?
> [I hope the brackets indicate how my mind groups these
functions).
> Would it be just as worthwhile to the student to teach
instead.
> (sec and its cofunction csc), and tan???
I think the reason for (sin/cos) and (tan) is that these are
the most
often used of the six in applied mathematics. Sin and cos are
the daily
bread for every graphics programmer, who has to calculate
pixel
coordinates on a screen. Or when it comes to Fourier analysis
or
anything else having to do with complex
Tan, on the other hand, is essential for calculating the
gradient of a
curve (the slope of a hill on the road signs is given in tan
as well).
I cant remember anything in applied math requiring sec, csc
or cot.
Anyone?
Steffen