Trigonometric Functions
Jordan C. Hack
Thanks for the insights into my previous questions; there is still one
unanswered question I have:
When graphing a trigonometric function, is it true that 1 on the y-axis,
and 1 on the x-axis should be the same distance from the origin? It
seems that in most texts I've read, and teachers in secondary school
don't follow this. Graphing y=sin(x) for example, they will put 1 and
-1 on the y-axis, and then (pi/2), pi, ... on the x-axis, but the thing
I've noticed is that 1, 2, 3,... on the y-axis is ticked off in roughly
the same intervals as (pi/2), pi, .... is on the x-axis. 1 is not near
(pi/2), just as 2 is nowhere near pi. Graphing sin(x) on a plane marked
in this way will lead to a sine curve that is much "taller" than it
should be (because (pi/2) does not equal one; it equals 1.5707....).
I've noticed that this "reckless" unit-marking on axes is usually
cleared up when two functions like f(x)=x, g(x)=sin(x) are shown on the
same graph to illustrate that sin(x) approx. = x, for x near 0.
Any comments on this?
noj...@here.there.ca
>Thanks for the insights into my previous questions; there is still one
>unanswered question I have:
>
>When graphing a trigonometric function, is it true that 1 on the y-axis,
>and 1 on the x-axis should be the same distance from the origin?
Not *necessarily*. It is just that people set arbitrary standards and
thus "see" the same things when having disussion. What *is* necessary
is that all the y-axis units have the same length, and all the x-axis
units have the same length. IMHO.
Jordan C. Hack
Ok, so if all the y-axis units have the same length, and all the x-axis
units have the same length, but someone chooses to make all the y-axis
units a little bit longer than the x-axis units, then the graphs of
their functions are going to be different than somebody elses.
Shouldn't pi units on y-axis represent the same distance from the origin
as does pi units on the x-axis? If not, then I'll choose a distance of
4 kilometers between units on my y-axis (ie. 4KM between 1 and 2, 4KM
between 2 and 3, ...), and for the x-axis, I'll choose a distance of 1cm
between units (so the distance from 0 to pi on the x-axis will be a
little over 1.5cm). Now when I graph f(x)=sin(x), it's going to look a
little bit taller than most peoples' graphs.
TRM
Jordan C. Hack wrote:
> Shouldn't pi units on y-axis represent the same distance from the origin
> as does pi units on the x-axis? If not, then I'll choose a distance of
> 4 kilometers between units on my y-axis (ie. 4KM between 1 and 2, 4KM
> between 2 and 3, ...), and for the x-axis, I'll choose a distance of 1cm
> between units (so the distance from 0 to pi on the x-axis will be a
> little over 1.5cm). Now when I graph f(x)=sin(x), it's going to look a
> little bit taller than most peoples' graphs.
If you can get that on a blackboard, more power to you.
If you're graphing by hand on the board, you need to make it fit. Graph
y = sin (x/2). amplitude 1 and wavelength 4pi, 12.something. It won't
fit unless you compress the lateral dimension.
Also. drawing the curve is difficult. I do one size well and look silly
if I try to change it so I draw my curve and then label the axes and
tick off the important points.
Terry Moore
In article <335D66...@undergrad.math.uwaterloo.ca>, "Jordan C. Hack"
<jch...@undergrad.math.uwaterloo.ca> wrote:
> When graphing a trigonometric function, is it true that 1 on the y-axis,
> and 1 on the x-axis should be the same distance from the origin?
Not necessarily. In applications of maths, it is useful to be able
to draw different scales for the domain and range of a function.
This allows you to fill the area available so as to see the greatest
possible detail. In many applications there is no standard scale.
For example if you want the relationship between people's
heights and weights they are not even in the same units. Changing
the unit on either axis corresponds to a change of scale.
However, if you want to verify that the gradient of the
function f(x) = sin(x) lies between -1 and 1, then using
the same scale is advantageous.
Terry Moore, Statistics Department, Massey University, New Zealand.
Imagine a person with a gift of ridicule [He might say] First that a
negative quantity has no logarithm; secondly that a negative quantity has
no square root; thirdly that the first non-existent is to the second as the
circumference of a circle is to the diameter. Augustus de Morgan
Justin Hahn
Jordan C. Hack (jch...@undergrad.math.uwaterloo.ca) wrote:
: Thanks for the insights into my previous questions; there is still one
: unanswered question I have:
: When graphing a trigonometric function, is it true that 1 on the y-axis,
: and 1 on the x-axis should be the same distance from the origin? It
: seems that in most texts I've read, and teachers in secondary school
: don't follow this. Graphing y=sin(x) for example, they will put 1 and
: -1 on the y-axis, and then (pi/2), pi, ... on the x-axis, but the thing
: I've noticed is that 1, 2, 3,... on the y-axis is ticked off in roughly
: the same intervals as (pi/2), pi, .... is on the x-axis. 1 is not near
: (pi/2), just as 2 is nowhere near pi. Graphing sin(x) on a plane marked
: in this way will lead to a sine curve that is much "taller" than it
: should be (because (pi/2) does not equal one; it equals 1.5707....).
: I've noticed that this "reckless" unit-marking on axes is usually
: cleared up when two functions like f(x)=x, g(x)=sin(x) are shown on the
: same graph to illustrate that sin(x) approx. = x, for x near 0.
: Any comments on this?
Marking of axes and the actually distance on the axes are not the
same. further there is no law that we must use the same scales on either
axis. anyone familiar with u-substitution in calculus (should be virtually
everyone) would (or should) realize that what that is is essentially a
rescaling of an axis. So no, you do not need to have axes which are labeled,
or even scaled, identically. It may be convenient to do so, but it also might
not.
--
- justin
"I hurt myself today to see if I still feel. Focused on the pain,
the only thing thats real." -NIN, "Hurt"
--
Jordan C. Hack
Ok, so if we both graphed f(x)=sin(x), then my graph could easily look
like sin(2x) or sin(3x) compared to your graph, depending on how I label
my axes, right? If we both label our x-axes identically, but I choose
to put 3cm between 0 and 1 on my y-axis and you only put 1cm between 0
and 1, then we'll have different looking graphs.
In light of this, it must be true that given a picture of a sine curve
(without the axes labelled), it is impossible to tell whether the
function is sin(x), 3sin(x), sin(10x), 400sin(.5x), .5sin(3000x), or any
other such variation..... is this correct?
John
In article <33628B...@undergrad.math.uwaterloo.ca>,
jch...@undergrad.math.uwaterloo.ca says...
If shown a plot of u vs A*sin(u) with no scales on the two axes, you
can mark off u = k*pi for any k. But since u = n*x for any n you can't
find x without a hint. You also can't tell the magnitude of A.
You can tell the sign of A with no hint except the plot.
How? (Hint: smoke and mirrors)
I haven't read all of this thread so the following might be
sententious, but I'm no virgin there.
It seems that you may be troubled by unequal x and y scales. If we
were laying out building lots, it would be annoying to have different
scales. We want feet north to south and feet east to west. Then the
drawing will be _similar_ to the actual dirt lot. (I mean similar in
the exact mathematical sense not in the sense that Australia is
similar to the US.) So we probably agree that building lots and
mechanical drawings should usually have equal scales.
In physics, it's often of interest to plot the temperature of an
object versus time. How would it be best to make the axes equal?
Oh, you ask, but what object?
EXACTLY!! It depends on the situation! The purpose of a graph,
usually, is to convey the most information in the clearest fashion
given the space on the paper.
So the temperature of a cup of hot coffee vs time would warrant a
temperature scale of say 50 C to 100 C and a time scale of 0 to maybe
200 minutes. I leave it as an exercise to establish reasonable scales
for the temperature of the sun from now until extinction in 5 or 10
billion years.
Most scientific and mathematical graphs fit in the second category.
More information can be transmitted by well chosen scales and those
scales will often be very different.
Regards,
John
--
Next scheduled rant concerns log vs log plots. Due out in 1999.
Brian Hutchings
In a previous article, jeh...@bu.edu (Justin Hahn) says:
not only that, but your "radian" axis can be labelled as *being*
in units of pis, as opposed to the redundancy of 0pi, pi/2, pi etc.;
conversely, your circumferential measure can be rational (or units) and
your radius can be transcendental (or piths .-)
>axis. anyone familiar with u-substitution in calculus (should be virtually
>everyone) would (or should) realize that what that is is essentially a
>rescaling of an axis. So no, you do not need to have axes which are labeled,
>or even scaled, identically. It may be convenient to do so, but it also might
--
Brian Hutchings, Living Space Programs, Santa Monica College
Time is the only dimension. --Bucky Fuller
TRM
Jordan C. Hack wrote:
> In light of this, it must be true that given a picture of a sine curve
> (without the axes labelled), it is impossible to tell whether the
> function is sin(x), 3sin(x), sin(10x), 400sin(.5x), .5sin(3000x), or any
> other such variation..... is this correct?
I made a single sine curve in WordPerfect. When I need a diagram, I
just label the axes appropriately. BTW, cos curve is just a lateral
shift.
tony richards
>
>In light of this, it must be true that given a picture of a sine curve
>(without the axes labelled), it is impossible to tell whether the
>function is sin(x), 3sin(x), sin(10x), 400sin(.5x), .5sin(3000x), or any
>other such variation..... is this correct?
The answer is yes.
The point to make though, is that a 'picture' of anything can be
misleading, unless a scale is marked on it.
So, its inportant to label and mark off some major scale intervals on
each axis of any graph you draw, so that
an uninformed viewer can draw the correct conclusions from it.
That 'uninformed' viewer may be your examiner.
--
Tony Richards 'I think, therefore I am confused'
Rutherford Appleton Lab '
UK '