[HN Gopher] Stacking triangles for fun and profit
___________________________________________________________________
Stacking triangles for fun and profit
Author : olooney
Score : 146 points
Date : 2024-04-11 12:57 UTC (10 hours ago)
(HTM) web link (www.oranlooney.com)
(TXT) w3m dump (www.oranlooney.com)
| jihadjihad wrote:
| Maybe I just got lucky, but that's pretty much exactly how I
| remember being taught in high school calculus class. No ODEs or
| anything of course, but the teacher started from the geometry,
| then moved to the trig functions and identities, and finally to
| derivatives and Taylor / Maclaurin series.
|
| It's a good post, and I agree--there is just no hope in getting
| students to develop any kind of intuition in math without
| starting from something really simple like the geometry of the
| problem. Plus, it's way more fun!
| darkwater wrote:
| I think fun in learning comes mostly from the "intuition
| reward". If you understand and _feel_ some abstract concept in
| an intuitive manner, you will feel happy and you will probably
| want more of that. So, for teachers finding ways to teach
| concepts that can trigger students intuition, in my experience
| as a student, will get more engaged students.
| Ensorceled wrote:
| I have my old high school calculus text book and that is
| definitely how trig was introduced. The proof of Pythagorean
| Theorem using sin and cos was literally part of my undergrad
| CALC101 (I remember because my high school teacher demonstrated
| it for fun so it was a rehash)
| lupire wrote:
| When were you in Calc 101?
|
| Before or after 2009?
|
| https://forumgeom.fau.edu/FG2009volume9/FG200925.pdf
| Ensorceled wrote:
| Before. It was clearly circular since the identity was
| assumed, the calc prof might have acknowledged this at the
| time.
| gedy wrote:
| My math teacher was teaching us Trigonometry without ever
| showing the unit circle, and the class was completely
| befuddling to me and other students. This was pre-internet, and
| I stumbled across the unit circle in a book and the meaning of
| sine, cosine, etc. on it - and was like a huge revelation to me
| and others.
|
| I've enjoyed trig since, and one of the few math topics I
| continue to use to this day.
| bbitmaster wrote:
| I remember just learning to code at a young age, but not
| knowing trig or seeing how it applied because I only knew it
| as being about triangles.
|
| I had learned to use some game graphics library (specifically
| DJGPP with Allegro, coding in C) and I was trying to make a
| 2d spaceship game, think like asteroids, where you can turn
| the ship and apply thrust. I couldn't figure out how to take
| the angle the ship was facing and get a direction to move.
| Basically I needed to understand the unit circle. I
| eventually found what I needed probably on some old internet
| forums. From that point on, trigonometry really started
| making sense to me.
| phkahler wrote:
| >> What struck me as odd when I was an undergraduate, and still
| strikes me to this day, is that none of these are the obvious
| trigonometric definitions about the opposite and adjacent sides
| of a right triangle.
|
| I had the less common experience of learning some graphics
| programming prior to taking a trig class in school. "How do I
| draw a circle?" You sweep an angle from 0 to 360 (or 2*pi) and
| use sin() and cos() to get points on the unit circle, then
| multiply by your radius and plot them. For me the notion of sin
| and cos being coordinates of points on a unit circle was natural.
| Later when I had trig class it seemed really weird to define
| these functions as ratios of sides of a triangle. In particular
| you never talk about a side length as being negative. So no, it's
| not universal that the triangle definitions seem more obvious or
| make more intuitive sense.
|
| I think this is important for people to understand. What seems
| easy or natural can depend on how your particular tree of
| knowledge was constructed.
| vlowrian wrote:
| In fact, sine and cosine never really made sense to me, until
| one day, I saw an animation similar to this one:
| https://youtu.be/Q55T6LeTvsA?si=7fiBxFVMu67tb3NZ (without the
| overly dramatic soundtrack).
|
| Before that it was just some weird function I needed to
| calculate angles in triangles...
| philomath_mn wrote:
| That's too bad -- I am pretty sure my trig class introduced
| unit circles and the triangle relationships at around the same
| time.
|
| The unit circle with key points labeled is still my go-to
| doodle; it is a beautiful set of concepts.
| Sharlin wrote:
| I remember wondering what these strange "sin" and "cos"
| functions were in Delphi 2 or something - it was ~1997 and I
| couldn't just google it - and tried to find a pattern in the
| seemingly arbitrary output values. I don't think trying to plot
| them as y=f(x) crossed my mind, not at first anyway, but
| instead I somehow ended up trying to plot x=k sin(t), y=k
| cos(t) and was amazed when I realized that they traced a circle
| on the screen. Only later I learned in school that they had
| anything to do with triangles. To me they were functions for
| emitting circle coordinates first and foremost.
| dmayle wrote:
| I started reading this, and the article is very approachable, but
| it's flawed from almost the very beginning.
|
| He uses the angle addition formulas to derive the Pythagorean
| theorem, but this derivation only works because he predefined
| sine and cosine as odd and even functions, which hasn't been
| proven
| lupire wrote:
| He did gloss over it, but odd cos and even sin is obvious as
| soon as you define signed angles (which and lengths (which he
| forgot to mention).
|
| Sin(t) clearly has the same sign as t. (That's why it's called
| "sin" ;-) )
| isolli wrote:
| My daughter (in 5th grade) is interested in learning advanced
| maths, so we looked at sine and cosine.
|
| Question: is there an easy, geometrical way to "show" that
| `cos(60o)=1/2` using the intuitive definition based on the
| trigonometric circle?
| orlandpm wrote:
| Yes! Draw an equilateral triangle with side length one. Then
| cut it in half and reason from there.
| sublinear wrote:
| Pretty sure we used to have posters of this and similar
| figures in middle school classrooms.
| gjm11 wrote:
| Yes.
|
| (I assume that by "the intuitive definition based on the
| trigonometric circle" you mean that (cos t, sin t) is the point
| at angle t on the unit circle. If you meant something else,
| then what follows may not be helpful.)
|
| APPROACH 1
|
| If you take the point (1,0) and rotate it anticlockwise through
| 60o about the origin, you get (cos 60o, sin 60o), by
| definition. Look at the triangle formed by (0,0), (1,0), and
| (cos 60o, sin 60o). The angle at the (0,0) vertex is 60o. The
| other two angles are equal because the triangle is isosceles
| (rotating a line segment doesn't change its length) so they
| must also be 60o since the angles in a triangle add up to 180o.
|
| So now drop a perpendicular from (cos 60o, sin 60o) down to the
| horizontal side of the triangle; it lands at (cos 60o, 0), so
| cos 60o is the distance from (0,0) to that point, and 1-cos 60o
| is the distance from (1,0) to that point. But those distances
| are the same, by symmetry, and now we're done.
|
| APPROACH 2
|
| Draw the unit circle and inscribe a regular hexagon in it with
| (1,0) as one vertex. Divide it up into six triangles by drawing
| radii. As above, these are equilateral triangles.
|
| So now look at the horizontal separation between the opposite
| points (-1,0) and (1,0). Obviously the distance is 2 units. But
| we can split it up a different way: we have a horizontal
| distance 1-cos 60o from either of those to the next point
| around the circle, then 1 from that to the next point, then
| 1-cos 60o back to the opposite point. So 2 = (1-cos
| 60o)+1+(1-cos 60o) and again we're done.
|
| (But I think any proof of this particular thing is going to be
| basically about equilateral triangles more than it's about
| circles. It might be easier first to show that for angles
| between 0 and 90o the trig functions have their "traditional"
| definition in terms of right-angled triangles, and then bisect
| an equilateral triangle.)
| adrian_b wrote:
| If you look at a regular hexagon inscribed in a circle (i.e.
| whose vertices are determined by 6 rotations of 60 degrees
| whose end point returns to the starting position), the length
| of one edge is equal to the radius (this could be demonstrated
| based on symmetry, because in the triangles formed by 1 edge
| and 2 radii all angles are equal, therefore also all edges).
|
| The cosine of 60 degrees is one half of the edge divided by the
| radius, i.e. divided by the edge, therefore it is 1/2.
| pvg wrote:
| As a side comment to sibling comments with various approaches,
| there's a handy bit of intuition/technique to this that's
| useful for 5th to 105th graders - when you notice 'useful'
| angles in a plane geometry problem, make the thing they suggest
| out of them. So, 60 deg angle is a piece of an equilateral
| triangle - finish the whole triangle and see if you get any
| ideas. The unit circle sin/cos thing teaches you all sorts of
| stuff about right-angled triangles. See a problem without a
| right angled triangle? Draw a line to add one, etc.
| lupire wrote:
| Useful angles are the only exact angles that really exist :-)
|
| Everything else is a brutish approximation.
| pvg wrote:
| Of course, 'useful' and 'brutish' can be stretched a bit.
| Maths olympiad problem (from right around 5th grade, to
| boot):
|
| Trisect a 36deg angle.
| gertrunde wrote:
| > Now, the line EC is perpendicular to AB, and the line BC is
| perpendicular is AC, so the angle [?]BCE is the same as the angle
| [?]CAD which we called a.
|
| I'm hoping that's a typo, and should be:
|
| > Now, the line EC is perpendicular to _AD_ , and...
|
| (Edit: although I vaguely recall in school doing that bit as the
| sum of the angles around point C)
| Tainnor wrote:
| Yes, I'm pretty sure you're correct, I did stumble over this
| too.
| gjm11 wrote:
| I don't agree with the perspective taken here; I don't see any
| reason to think that the definition in terms of triangles is the
| simplest or most natural, and I suspect the author feels that way
| just because it's how they were first taught about
| trigonometrical functions.
|
| Of course I _do_ agree that definitions in terms of power series
| and differential equations are less natural and require heavier
| mathematical machinery.
|
| But: the "right-angled triangle" definitions have the _severe_
| drawback of only applying to a restricted range of angles, and
| their relationship to those higher-tech definitions is more
| indirect than necessary.
|
| Instead, I claim that the One True Definition of the trig
| functions is: if you start with the point (1,0) and rotate it
| through an angle t (anticlockwise, which is conventionally the
| "positive" direction in maths) about (0,0), then the point where
| it ends up is (cos t, sin t).
|
| This is just as simple, and just as geometrical, as the
| "triangle" definition. It leads directly to the differential-
| equation characterization (which in turn leads easily to the
| power series). For angles between 0 and pi/2, it's obviously
| equivalent to the "triangle" definition.
|
| Can you get the addition theorems easily from this definition?
| Yes, and (again) without the severe drawback of only applying
| when all the angles involved are between 0 and pi/2 as for the
| "triangle" definition.
|
| Start with a diagram showing the points (0,0), (cos t, 0), (0,
| sin t), (cos t, sin t). Now rotate the whole thing through an
| angle u about the origin. Obviously (0,0) stays where it is. (cos
| t, 0) just goes to cos t times (cos u, sin u), i.e., to (cos t
| cos u, cos t sin u). If you turn your head through 90 degrees it
| becomes clear that (0, sin t) similarly goes to (- sin t cos u,
| sin t sin u). And of course (cos t, sin t) goes to (cos t+u, sin
| t+u) since we have rotated it through an angle t and then through
| an angle u.
|
| And now we're done, because the "vector addition" property the
| original diagram had remains true after rotation, so adding (cos
| t cos u, cos t sin u) to (- sin t cos u, sin t sin u) has to give
| you (cos t+u, sin t+u). And that's exactly the addition theorems.
|
| (That may be hard to follow with no diagrams, but I think it's
| easier to follow than the triangle-stacking proof would be
| without diagrams, and with diagrams everything is pretty
| transparent.)
|
| The right-angled-triangle definitions are traditional because
| historically trigonometry came before coordinates. But now we
| have coordinates and generally learn about them before we learn
| trigonometry, and at that point the point-on-the-unit-circle
| definitions are simpler, more general, and better suited for
| proving other things.
| enizor2 wrote:
| Could you expand on getting the differential equation from your
| definition? I don't really where to start from it.
| outop wrote:
| Draw a line of length 1 which makes an angle of p with the
| positive x axis. The x and y coordinates of a, the point at
| the end of the line give cos(p) and sin(p).
|
| Now think about what happens if you increase p by a tiny bit.
| a moves tangentially. (This works very similar to how the
| tangent to a curve gives the change in y for a change in x.)
| So the vector of cos'(p), sin'(p) is given by a vector
| starting from a, at a right angle to 0a, and pointing in the
| positive direction.
|
| Since the point a moves through 2pi distance while p goes
| from 0 to 2pi (the definition of measuring angles in radians)
| the speed of the point is 1, and so the vector of derivatives
| has length 1.
|
| You can check easily that this makes cos'(p) = -sin(p) and
| sin'(p) = cos(p).
| gjm11 wrote:
| Yup!
|
| (If I were trying to present this stuff in a maximally-
| elegant order without _too_ much regard for what order
| human brains like to learn things in, the order of things
| would be: complex numbers, calculus, trigonometry. Then we
| define something that we might initially call e(t) to
| satisfy the differential equation de /dt = ie, and observe
| that having e and e' at right angles means that |e| remains
| constant, which means that |e'| also remains constant, so
| if we start with e(0)=1 then we have a point moving at unit
| speed around the unit circle, etc. Keep the linkage between
| the geometrical and formal points of view there at all
| times. But I suspect this wouldn't be great paedagogically
| for the majority of students.)
| lupire wrote:
| If you allow negative side lengths, as you should, much of
| geometry is unified and simplified, such as this situation.
|
| Negative (and imaginary) measurements make math better-behaved.
| (See also: quantum mechanics.)
|
| Rectangular coordinates are suboptimal because they are
| arbitrary in a way that hides some of the symmetry of
| mathematics.
|
| Your vector argument looks like a lot of algebra noise "without
| a diagram" because it relies on... triangles... for intuitive
| justification.
| tromp wrote:
| > Instead of adding two separate angles a and b, we'll use th
| and -th.
|
| It's not clear that the geometric proof based on pictures
| with positive angles a and b also applies to a negative b. I
| think one should at least provide a separate picture of that
| case...
| Tainnor wrote:
| > If you allow negative side lengths, as you should [...]
|
| I mean, the blog post was about a pedagogically better suited
| construction of trig functions. If we allow negative side
| lengths, it's not really going to be intuitive anymore. At
| this point, you might just as well use the power series
| definition (or the one based on the complex exp function)
| since it makes things easy to prove.
| lupire wrote:
| Imaginary lengths may be unintuitive but negative is pretty
| simple.
|
| The idea of negative measurements (meaning "opposite
| direction along a line") as is well understood before
| children study geometry. Kindergarteners learn "left" vs
| "right" and "forward" vs "backward".
|
| I agree with other poster about the value of showing
| negative and positive lengths visually in the same diagram.
| Tainnor wrote:
| Magnitudes are intuitionally directionless (hence why
| norms and measures are nonnegative reals). Of course, you
| can extend notions suitably and make it work, but I
| really don't think that this is terribly intuitive. (Of
| course, different people will find different things
| intuitive.)
| jvanderbot wrote:
| The rotating angle suggests immediately the domain and
| definition of Sin(x), Cos(x) using simple projections. And,
| huuuuge bonus, that's a right triangle.
|
| I just think lesson zero in the above article should have
| been these projections, and a simple "From this all angles
| can be defined as right triangles in your preferred
| coordinate system", and you're off to the races in any old
| direction.
|
| This is _not_ how I was taught it, and I 'm retrospectively
| upset.
| Tainnor wrote:
| I agree that the triangle-based definition starts breaking down
| once you get to angles greater than pi/2. I was told in school
| that "there are no right triangles with such angles, but just
| imagine that there were" and was then shown some weird
| (supposedly suggestive) diagrams. I found this unsatisfying and
| moreover hard to remember.
|
| Of course, you could always just define sin and cos for acute
| angles only and then extend the definition with trig
| identities, but that seems rather unmotivated too.
| justinpombrio wrote:
| You phrase this as a disagreement, but in my mind this is
| complementary.
|
| The point of the post is that you should define things in terms
| of what you care about, and _then_ prove stuff about it. The
| sum: Sum from n=0 to infinity of (-1)^n /
| (2n+1)! * x^(2n+1)
|
| isn't something you should _start_ with, it should be the
| _punchline_. Instead, you should start with angles (the thing
| you care about), then prove that they behave the same as that
| sum (what an incredible claim!).
|
| The post proposed "angles of the acute corners of right
| triangles" as the starting point. You've argued very well that
| "angles in the unit circle relative to (1, 0)" is a better
| starting point. Pedagogically, I think it's a wonderful
| starting point:
|
| "Let's talk about angles. Just angles, nothing else. When you
| look at angles of actual objects, the side lengths are all
| different, which complicates matters. Since all we care about
| is the angle itself, not the side lengths, let's make all side
| lengths equal to 1. Etc. etc."
| jessriedel wrote:
| It wasn't a historical _accident_ that triangles came before
| coordinates. Coordinates are more abstract than triangles quite
| generically for humans. (When we teach children numbers, we
| start with the natural numbers and then later introduce
| negative numbers.)
|
| I claim it's only because you and I have internalized them so
| well that they both seem intuitive, which is what allows you to
| prefer the coordinates approach due to its greater generality
| to negative angles.
|
| In any case, as another commenter said, I think you basically
| agree with the author's main point and are disagreeing with a
| minor point (which is to some extent a manner of taste).
| jerf wrote:
| This is another way of looking at the concept of Kolmogorov
| Complexity and relating it to real life... which thing you
| consider the most "natural" encoding of these concepts, and
| thus the "shortest", can depend very heavily on your "native
| mathematical language". One not even reach out to strange
| hypothetical aliens who think utterly different than us.
| Multiple people all firmly raised in the modern conventional
| human mathematical landscape can vary, as is seen right here.
|
| Though it does occur to me to wonder what an alien in a very
| obviously hyperbolic universe would consider the most
| natural. Or one of the beings that lives in Greg Egan's
| universe with two time and two space dimensions.
| onedognight wrote:
| The only reason the author's definition doesn't apply to larger
| and smaller angles is that they explicitly considered the
| angles <ABC and <CBA to be equal rather than negatives of each
| other. That was a surprisingly odd oversight given that they
| immediately start talking about negative angles.
| woopwoop wrote:
| I guess it's all in the way you look at things. I would say that
| the addition formulae for sine and cosine are more weird and
| technical than the Banach fixed point theorem, which I would say
| is much more fundamental.
| enizor2 wrote:
| I do not understand this consideration: > By considering a
| triangle with hypotenuse 1 and a very small "opposite" side, it's
| not hard to see geometrically that sin(x)[?]x and cos(h)=x when x
| is small
|
| I fail to see how you can "see" finer than sin(h) -> 0 & cos(h)
| -> 1
|
| From the limit definitions you actually need :
|
| * (1-cos(h)) / h -> 0
|
| * sin(h)/h -> 1
|
| (which correspond to the derivatives at 0).
| lupire wrote:
| Your limit definition is the same as the part you quoted, so
| it's not clear what your question is. I also don't see what you
| are quoting.
|
| Curvature is inverse of radius.
|
| Decreasing angle is equivalent to increasing radius, and this
| decreasing curvature. This, as angle decreases, the curve
| becomes close to a straight line, and that straight line
| approaches a vertical line.
|
| As usual, 3B1B created a quintessential visualization and
| explanation.https://m.youtube.com/watch?v=S0_qX4VJhMQ
| enizor2 wrote:
| I quote the second paragraph of the Derivatives section.
| (which was edited to a better, but not yet enough, sin(h)[?]h
| and cos(h)[?]1 when h is close to zero).
|
| I perfectly understand that around 0, sin(x) ~ x and cos(x) =
| 1 + o(x) but it isn't obvious geometrically, unlike what the
| article implies.
|
| From my point of view, increasing radius / decreasing
| curvature only gets you sin(x) -> 0 ; cos(x) -> 1, but that
| isn't enough to obtain the derivatives.
|
| I found a geometric proof in [1] but that part is the longest
| and hardest of the page. I was wondering whether the author
| found a clearer way to express is.
|
| [1] https://www.mathsisfun.com/calculus/derivatives-trig-
| proof.h...
|
| EDIT: after looking at 3B1B's video, the "small" triangle
| d(sinTh) by dTh figure would be a better way to explain the
| derivative, rather than an "not hard to see geometrically"
| approximation that isn't enough to conclude.
| philsnow wrote:
| That one line was the part that stood out to me the most as
| well, but:
|
| If you zoom in sufficiently at x = 0, f(x) = sin(x) looks
| indistinguishable from f(x) = x, whereas g(x) = cos(x) looks
| indistinguishable from g(x) = 1.
|
| (also, sin(x) is negative approaching 0 from the left and
| positive approaching 0 from the right)
| mckn1ght wrote:
| > none of these are the obvious trigonometric definitions about
| the opposite and adjacent sides of a right triangle
|
| Am I just misunderstanding something about this articles
| motivation? I'm pretty sure I learned the unit circle in high
| school trig, possibly even 7th grade geometry although my memory
| that far back is fuzzier; but we did lots of geometric
| constructions with straightedge and compass, and did basic
| geometric proofs using complementary angles etc, and my teacher
| was obsessed with triangles. I didn't learn about series until
| Calc 2 in early undergrad.
|
| I still use the unit circle to reconstruct various trig
| properties from memory.
| lupire wrote:
| His claim is that sin and cos in calculus are often introduced
| _independently_ of geometry, as math for engineers is non
| rigorous, and the connection to triangles is a magical
| coincidence.
|
| See also "Early vs Late Transcendentals" in calculus pedagogy.
| munchler wrote:
| I agree, and the linked Wikipedia page starts out with the
| obvious definitions as well: In mathematics,
| sine and cosine are trigonometric functions of an angle. The
| sine and cosine of an acute angle are defined in the context of
| a right triangle: for the specified angle, its sine is the
| ratio of the length of the side that is opposite that angle to
| the length of the longest side of the triangle (the
| hypotenuse), and the cosine is the ratio of the length of the
| adjacent leg to that of the hypotenuse.
|
| The author's claim that other definitions are the "most common
| starting points" seems like a straw man.
| vanderZwan wrote:
| I wonder if the author of this article would like Norman J.
| Wildberger's work on rational trigonometry[0], which also argues
| that angles and unit circles are the wrong starting point for
| defining triangles.
|
| [0]
| https://en.wikipedia.org/wiki/Divine_Proportions:_Rational_T...
| Tainnor wrote:
| The article initially points out that in mathematics, there are
| often equivalent definitions, that each have their own benefits
| and drawbacks. I think the author could have just written "I have
| found an alternative approach" instead of "I have found a better
| approach".
|
| As others have noted here, the geometric argument only works
| "intuitively" for acute angles and the functions have to be
| explicitly extended. Still, I hadn't seen this proof of the angle
| addition formula yet, and I found it neat.
|
| From a point of view of formalisation, a power series based
| approach (either directly or via the complex exp function) as
| traditionally used is probably better, because going into
| analysis (especially complex analysis), you're going to need
| power series anyway. Geometry meanwhile is intuitive to us but
| you'd have to encode a bunch of Euclidean axioms and theorems
| beforehand, which you might not otherwise use. Also, for better
| or worse, many mathematicians aren't really taught axiomatic
| geometry (I wasn't at least).
| tel wrote:
| I think there's a bit of a straw man here pointing at the series
| definitions as being applied as the "intuitive" sense for sin and
| cos.
|
| Instead, I find that the intuition that's sought is more to start
| by seeing exp(it) as being a generator of complex rotation---a
| tremendously beautiful and parsimonious bit of theory---and then
| seeing sin and cos as being 1-dimensional coordinate projections
| of that.
|
| Then the series definitions are just cute ways of deriving that
| relationship formally.
|
| Circles over triangles.
| dhosek wrote:
| One of the nicest definitions of the six basic trig functions
| involves drawing an ray from the origin of a unit circle with its
| center at the origin.1 Where the ray intersects the circle at
| point _A_ draw a vertical line perpendicular to the _x_ axis. The
| height of the line segment from _A_ to the _x_ axis will be the
| sine of the angle between the ray and the _x_ axis. The length of
| the line segment from the origin to where your vertical line hits
| the _x_ axis will be the cosine.
|
| Now, draw a tangent line perpendicular to the _x_ axis and find
| the point _B_ where your ray intersects the tangent line. The
| length of the segment from the origin to _B_ will be the secant,
| the length of the segment from _B_ to the _x_ axis will be the
| tangent.2
|
| Finally, draw the tangent line parallel to the _x_ axis and find
| the intersection of the ray with that line at _C_. The length of
| the segment from _C_ to the origin will be the cosecant and the
| segment from the _y_ axis to _C_ will be the cotangent.
|
| You can use your basic trig identities and knowledge of similar
| triangles to verify the relationships between the functions and
| the triangles. Angles outside the first quadrant will give signed
| values that make sense if you consider segments going down or
| left to be negative (but down _and_ left is positive).
|
| [?]
|
| 1. I'm making reference to cartesian coordinates strictly for the
| sake of convenience since I'm using only words to describe a
| diagram.
|
| 2. I'm doing this from memory and really hoping I'm not mixing up
| the tangent, cotangent, secant and cosecant
| dhosek wrote:
| I think I have somewhere a nice drawing of this I did in
| Illustrator back in grad school.
| salahalzoobi wrote:
| ok
| personjerry wrote:
| Also fun exercise:
|
| Go through Euclid's Elements (i.e.
| http://aleph0.clarku.edu/~djoyce/elements/bookI/bookI.html)
|
| Read the definitions and then prove all the postulates yourself,
| in order.
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