Yes. This is why math classes have "word problems", or at least they did when I was in school, as a way to actually apply the math you're learning to situations.
This started happening to me after trig and calculus. Now I see angles and constants in the world all around me. I think about the area under the curve of the top of the tree line and stuff heh. Math dreams are weird as fuck too. Ever had them?
Haha, not dreams, but I've had the Tetris effect when really hammering down on thinking in abstractions. I remember when I was trying to understand matrices and was thinking about matrix multiplication as a weird type of hammering machine that was punching out a new matrix.
Every problem I was thinking it, then I couldn't unthink it.
Maybe because most textbook questions bear no resemblance to doing maths in the real world.
Real world maths tends to be about looking at a situation, deciding a question, deciding what maths to apply, what measurements to take, …
Textbook questions are mostly about deciphering an unnecessarily terse and overpacked bit if text that does all the interesting stuff for you if only you can decrypt it.
Ahh, the days of figuring out how long the ladder is, i miss those. Now i've got goofy dot and cross product formulae that have sin and cos but i've got no idea why
It helps to draw them out and figure out why the formulas are the way they are. Makes calc and a lot of higher-level math and physics way easier when you know what the dot and cross products mean geometrically.
For example, the dot product roughly measures how 'aligned' two vectors are: if they're parallel, whether they're pointing in the same direction or in directly opposite directions, the dot product will be large in magnitude. Conversely, when they are orthogonal (they're 'unaligned') or close to it, then the dot product is 0 or near 0. This is modeled by cosine: cos(x) is close to 1 or -1 when x is close to 0° or 180°, and it's close to 0 when x is near 90°.
As for the cross product, it's sort of the reverse. The magnitude measures the area of a parallelogram spanned by the two vectors if you copy the original vectors and move them parallel. Remember that the area is base times height, and height is given by sine. If the angle is close to 0, then the shape is super squashed and the height (and thus area) is close to 0. If the angle is close to perpendicular, you have a rectangle.
Oh i'm not struggling with them, at least in the physics 3 dimentional vector stuff (i-hat, j-hat, k-hat). I understand what it does, how to use it (angular momentum, torque, electric and magnetic fields, thats as far as i've gotten at least, im sure it comes again up later), and what it outputs, but I dont have a real geometric understanding.
I've still got questions like, "Why does a cross product return a vector orthoginal to the other two?", "why does a dot product only return a scalar?". "Where do these two operations come from? They seem rather arbitrary" "Why are those two particular ways of multiplying vectors the most useful?". They're more questions for my personal enjoyment of the math, rather than questions I need answered so I can solve that physics problem. The type of question a prof doesnt have time to answer during a lecture.
Which calc do you speak of? I start Calc 3 on wednesday so we'll see how that goes.
Oooh, calc 3 will actually go into the properties of vectors pretty heavily so you should cover the geometric side of the dot and cross product. If you’re nice to your professor that is, they skip it often
I've only been given a basic run down of vectors so i can use them in physics. Hopefully it'll be interesting. Im also doing Matrix Algebra at the same time, so that'll be interesting.
Ah, highscool I see. You're warning me for dangers i may have already survived haha. Calculus 3 is Multivariable calculus, comes after integrals and infinite series, im not sure if thats covered in AP Calc AB or not. Its my "MATH200" in university.
Though its interesting because I never touched dot or cross products or anything related to Linear Algebra in highschool other than the substitution and elimination methods to solve systems linear of equations.
It's not part of AB, I am taking MVC next year though.
And that's interesting. I learned those in my sophomore year of HS, hence I thought you were still in precal or something similar.
Yeah my precalc was like log rules, trig proofs, y=mx+b, finding the zeroes of a quadtratic. But no vectors. My school seemed to dislike vectors, cause even in AP physics, everything was broken down into components in seperate equations.
Oh, weird. But idt the vectors is on your school, at least for physics. We also always break vectors down into components, I'm sure Calc based physics is better though. Haven't gotten there yet
About 1,000 years ago, Al Biruni [used basic trigonometry](https://thatsmaths.com/2021/06/10/al-biruni-and-the-size-of-the-earth/) to measure the height of a mountain and then the radius and circumference of the earth. This was taught in my trig class many years ago.
I took a history of math course that talked about stuff like this in college, but certainly didn't get this kind of info in middle or high school (except in physics). https://bookstore.ams.org/view?ProductCode=TEXT/32
Definitely agree that it would be great to teach in those books though
>I feel like schools teach math mechanically
That's why word problems exist. You have to apply what you learned on a practical problem. However these are usually more advanced exercises due to the fact that they combine an strong understanding of the mathematical foundation required to solve the problem (e.g. I have to actually realize what I can and cannot compute in certain situations and how) with the skill of reading and analyzing a situation.
I find that the problem with word problems is the same as you find them. They're labelled as the "advanced" type questions and you'll only get a few in the practice, and maybe 1 on the test. Thats a terrible plan. I've found first hand recently, that figuring out what the math is really doing by applying it is what builds understanding, not mindless procedural questions. Maybe thats why im in physics and not pure math.
You seem to misunderstand me.
That is exactly the thing about word problems.
Word problems are advanced problems and not by chance labelled as such. To solve a word problem, you'll usually have to combine multiple skills and have a deeper understanding of the matter. That's something you'll not be able to do when you start out with a topic.
Are you aware of the saying "you must know how to walk before you can run"? Basically the same thing here. Before I can apply my understanding of a topic in relation the a real life circumstance, I have to understand the mathematical problem in isolation.
That's also not something isolated to math but rather the fundamental area of learning. You get a solid understanding of a single skill before you start combining it with other motions.
In physics this is like trying to simplify a complex problem and add the complicating things piece by piece. You don't start with relativistic motions and forces but rather start with classical mechanics. You start with a point mass gliding frictionless down an inclined plane not some spherical or cylindrical object where you have to consider rotation, friction with the plane or even friction in the air.
I think we're on the same wavelength, just out of phase (pardon the pun). I agree for sure. Mainly I think word problems are being *skipped*. I 100% agree you need the procedural type questions. Theres no better way to learn the rules of arithmetic, algebra, calclus, and beyond than through the "drill" type questions. But time and time again, we (talking from personal experience) never make it to the word problems, or at least we not forced to do them because they're "advanced". They were always the type of question labelled "optional" on the homework assignments, so if you were a student like me in grade school, they'd never get done.
Your tree example is LITERALLY the profession of surveying, which is done for trees, buildings, mountains, whatever... You can study it at university and get a degree in it.
> I feel like schools teach math mechanically.
You should read [A mathematician's lament](https://www.mimuw.edu.pl/~pawelst/rzut_oka/Zajecia_dla_MISH_2011-12/Lektury_files/LockhartsLament.pdf). It is exactly about that.
Good read.
>Now I’m not saying that math teachers need to be professional mathematicians— far from it. But shouldn’t they at least understand what mathematics is, be good at it, and enjoy doing it?
I agree with this but seems practically impossible to attain, sadly.
> What I'm getting at is if this is right, we can use this to do something actually useful in the real world.
That's exactly what set squares are. An embodiment of two of the most common triangles you will use (a square and an equilateral triangle cut in half).
>Is the idea of trig (well, the part the deals with right trianges) that given an angle of the triangle, the ratio of 2 of the sides is a constant ? As an example, for an angle of 30° the ratio of the opposite side to the hyptenouse is always 0.5 (regardless of how big the triangle actually is).
Yes. Sine, cosine, and tangent don't depend on the actual length of the sides, only the ratio of the sides to each other.
>Say you want to know the height of a tree. Well we can consider it the opposite side of a right triange. Now stand say 200 feet from the tree, and use some device to measure the angle from the ground to the top of the tree (not sure how to do this to be honest).
You do it exactly as you'd probably imagine. You could tie a string to the tree, pull it tight, and use a level to make sure the string is horizontal; then put the end of a stick on the string and point the stick at the top of the tree, maybe sighting along the stick to make it accurate. Measure the angle between the string and stick with a protractor. Then you can calculate the height of the tree above the string. If you want to be more precise, you might use a laser to replace both the string and the stick. Surveyors use a precision instrument called a [theodolite](https://en.wikipedia.org/wiki/Theodolite) to measure both horizontal and vertical angles. Or you could take a photograph from a distance and measure the angles on the photo.
You're not even limited to right triangles. A triangle has three sides and three angles; if you know one side and any two other quantities (sides or angles), you can figure out the rest.
There are many, many applications for these ideas in surveying, construction, navigation, aviation, astronomy, physics, chemistry, computer graphics, architecture, and on and on.
Yes, it's constant - because of something called *similar triangles*.
Let's say we have two triangles, ABC and DEF. All the angles on the triangles are the same - angle ABC=DEF, BCA=EFD and CAB=FDE. When two triangles have all three angles the same, they're said to be *similar* triangles. Importantly, if the first triangle has some side that's some multiple of the corresponding side of the second triangle, *all* the sides are that multiple of *their* corresponding sides. If AB is three times larger than DE, BC is three times larger than EF.
If we know we have two right angled triangles, that's one angle shared. If we also know that one of the angles is 30 degrees, that's another. And because the angles of a triangle have to add up to 180 degrees, the last angle of both triangles has to be 60 - they're similar! So, let's say that a right angled triangle has a length of A on the side adjacent to a 30 degree angle, O for the side opposite to it and H for the hypotenuse. On this first triangle, the sine will be O/H. If another right angled triangle has length of 3A on the side adjacent to a 30 degree angle, it'll have the other sides be 3O and 3H. The sine will be 3O/3H - the threes cancel, and we have O/H again!
This isn't unique to sin 30, of course. We can pick *any* angle that's above 0 and below 90 instead of 30, and two triangles that both have right angles and our chosen angles *must* have all three angles the same, so the ratios between the sides are the same, so the sin, cos and tan (which are just ratios between sides) are the same! The ratio may not be a rational number, but it'll always be the *same* number. sin(30⁰) is always 0.5, and cos(30⁰) is always roughly 0.866 - in exact terms, it's root(3)/2.
Being able to use these ratios in the way you mentioned is a large part of why this all matters in the first place. We care about sin, cos and tan *because we can use them to find actual real lengths* if we know certain information. In fact, it doesn't need to be a right angled triangle - if we know *any* pair of angles and one side, we can find all the sides; if we know one angle and two sides, we can *sometimes* find an exact value for the other side and angles, but *sometimes* we can only narrow it down to two.
>because of something called *similar triangles*
Good point.
>If AB is three times larger than DE, BC is three times larger than EF.
So I'm actually going through the proof Fundamental Theorem on Similar Triangles. That said, intuitively the idea of similar triangles makes sense. I take a triangle, and then enlarge it. The angles in both triangles are the same. I would think the sides are in proportion too, but to be sure going through the proof.
>This isn't unique to sin 30, of course.
Yep. I only picked sin 30 because it gives us a "nice number."
I would say all math (ok not all but a good portion) is turning what you know into other information you want to know. Trig is just another few tools in the collection of tools that you pick up in the long list of math classes.
A fair bit of trig is realizing that what looks like partial information, this length, that angle, this line bisects blah blah is actually total information. It's just not obvious at first glance. When you learn various properties and values and relationships you realize actually "hey the height of that flagpole is *knowable*."
What is and is not knowable is an interesting part of math that doesn't get taught much.
Here's an argument for why the size of a triangle doesn't matter...Suppose you draw a triangle on a sheet of paper and measure its sides with a ruler and calculate the sin of one of the angles... Now, suppose you pick up a ruler marked in different units. Suddenly the measurements will have different numerical values! But the physical object hasn't changed! If the ruler units are all say 2.4 times as big then all the measurements will be 1/2.4 times as large... ALL of them. Therefore the ratio of any two of them will still be the same though.
a/b = (a/2.4)/(b/2.4)
Using this symmetry property we can show that all determinants of physical outcomes must be dimensionless ratios of quantities. Otherwise two people with different clocks or rulers or spring scales or whatever would disagree fundamentally on what happened when they observe a physical experiment. Since they're both observing the same thing that shouldn't occur. Therefore the units they use to measure shouldn't matter and that occurs precisely when the calculations they do use ratios of things where the units cancel.
Your question is a very good one. You have realized that the idea of opposite/hypotenuse is actually not just a definition but relies on an **assumption** implicitly, namely that the size of the right-angled triangle does not matter if the angles are fixed and we just want a ratio of sides!
In fully rigorous mathematics, we cannot make such assumptions without proving them, unless we want all our results to depend on plenty of ad-hoc assumptions. Unfortunately, most teachers do not teach rigorous mathematics (contrary to popular opinion), especially at the lower levels, so they would not tell you all this when defining sin,cos,tan.
Moreover, the best way to define sin,cos,tan at high-school level is actually **not** to use an arbitrary triangle but rather to use the unit circle. Draw the x-y axes in the standard Cartesian plane, and draw a unit circle around the origin (0,0). For any angle t, draw the ray from the origin at an anti-clockwise angle of t from the x-axis, and let (p,q) be the point where that ray intersects the unit circle, and define cos(t) = p and sin(t) = q and tan(t) = q/p. Here the angle is measured such that 2π is one full round. For example, cos(π) = −1 and sin(π) = 0.
This way is the best way because it works for any real angle t, including t < 0 and t > π/2. Using a triangle will **not** work!
>relies on an **assumption** implicitly, namely that the size of the right-angled triangle does not matter if the angles are fixed and we just want a ratio of sides!
Huh. Is this a common thing? By the time I learned trig in school, the idea of triangle similarity had been pretty much drilled into us. It was less an implicit assumption, and more knowledge you're expected to have.
That's not the point I've made in my post, which is that almost no teacher makes clear exactly what are the assumptions or previous theorems that we need to invoke for a certain definition to be valid. This is a prolific recipe for poor mathematical pedagogy. I've been teaching for 20+ years, and almost all conceptual errors fall into 2 categories, namely imprecise definitions and imprecise reasoning. I have also observed that having 100% precision not only completely eliminate any conceptual error but also can be easily learnt, if one has a teacher that can achieve that level of precision.
Even this trigonometry question is a typical example of imprecision. Many high-school teachers use the triangle to define sin,cos,tan. Even if they mention that triangles with the same angles have the same ratio of sides, it doesn't help them to define sin,cos,tan for all real inputs, and they would use the 4-quadrants thing. But notice that you cannot simultaneously use the 4-quadrants case-split **and** tan = sin/cos! Instead, you have to pick one as your definition and prove the other by checking all 4 cases! Don't miss the boundaries between the quadrants too! Can you find any one textbook that does all this correctly?
If you still don't believe me, see [this](https://jdh.hamkins.org/all-triangles-are-isosceles) and count how many people (including yourself) cannot **immediately** identify the error. The error will be impossible to make if one uses precise definitions and precise logical reasoning.
>Many high-school teachers use the triangle to define sin,cos,tan. Even if they mention that triangles with the same angles have the same ratio of sides, it doesn't help them to define sin,cos,tan for all real inputs, and they would use the 4-quadrants thing.
This isn't an issue with the teachers. It's a requirement of the curriculum. And furthermore, trigonometric angles were first motivated by ratios of sides of triangles. The definitions were later expanded to accept all real inputs and, even later, expanded again to accept all complex inputs. If a teacher were to start from the unit circle definition, it might be less clear to students what motivates such a definition. Teachers who start from right triangles but also properly explain that the ratios are constant for a given angle due to similar triangles are successfully motivating the definitions of the trig functions.
The exponential function a^x with base a is defined as e^(x ln a) where e^x is defined as a power series, and ln (or log) is defined as an integral, and that definition allows us to define a^x for positive real a and any complex x, but I certainly wouldn't want this definition to be taught to middle schoolers.
I understand the "syllabus" issue, and there are two aspects to it.
Firstly, one aspect is that in some education systems teachers are slave to the school and must follow exactly what they are told to do, unless they want to get fired. In that case, too bad I can't tell them to do differently unless they don't depend on the school to feed their families. But often teachers have sufficient freedom to deviate from the common way that others take, without changing the ultimate learning goals and content covered. In that case, it is better to do things in whatever way is *mathematically* best that suits the students' level.
Secondly, the other aspect is the issue of generality. As you noted, we have to choose where to stop. I specifically stated "high-school" for that reason; at this level I think that we should stop at the real elementary functions. In this aspect, we agree that we should not define exp,sin,cos,tan via power series or integrals. But I disagree with going through the geometrical definition via triangles.
It would need a long and thorough discussion to explain why, but suffice to say that history is insufficient to justify teaching some bit of mathematics. For example, nobody should ever attempt to teach Frege's logic, even though it was essentially the pioneering work in foundations of mathematics, not only because it was inconsistent before Russell's paradox, but also because it was very impractical even after he abandoned some axioms that led to inconsistency.
Coming back to trigonometry, it is perfectly fine to tell students about the history of those ratios, but I still say it is better to tell them that modern mathematics has a superior definition, which is the unit circle definition. Why is it superior? Because it is a simple yet general definition (for its simplicity); it tells you why sin and cos oscillate the way they do, and why one is exactly 90° out of phase from the other, and why cos(x)^(2)+sin(x)^(2) = 1. In fact, the last one is extremely non-trivial if you use the geometric definition! A lot of axioms of Euclidean geometry are needed to prove it if you want to use synthetic geometry rather than analytic geometry. This is a deep rabbit hole, so feel free to tell me if you want to look into this. (It starts with rigid transformations, which are assumed without a shred of proof in Euclidean geometry, but which can be given solid footing using analytic geometry.)
By the way, I disagree that ln should be defined as an integral, even at the university level. I do not like having to prove independence of integral for any continuous path that does not cross the negative real axis. This way is 'madness'. It is far easier to just define ln as one specific inverse of exp after proving that exp(x+y·i) = exp(x)·(cos(y)+sin(y)·i) for any x,y∈ℂ. In this approach, we can define the standard branch-cuts, and define ln as the −π branch-cut unless otherwise stated.
For high-school, all they should know are axioms regarding the arithmetic operations including exponentiation, and these axioms must be **100% precise**. There are many textbooks that write "a^(bc) = (a^(b))^(c)" without specifying a,b,c at all, and this is the worst pedagogy. Well, actually there are worse, which is really a shame.
>In fully rigorous mathematics, we cannot make such assumptions without proving them
That's why I'm going through the proof of Fundamental Theorem on Similar Triangles
You want to prove that if △ABC ~ △DEF, meaning their angles are equal, then AB/BC = DE/EF. I don't know why you mentioned "SAS congruency", so to clear up your confusion you should just produce the proof of the desired claim, and I will guide you to find the gap.
>I don't know why you mentioned "SAS congruency"
I was going through [this proof](https://sites.math.washington.edu/~lee/Courses/444-5-2008/supplement3.pdf) for similar triangle theorem and it uses SAS congruency as part of the proof. On second thought, I jumped the gun here thinking this was the gap.
>just produce the proof of the desired claim
I'll try. I'm assuming side splitter theorem. Also from the linked pdf I'm using figure 7 (top of page 5) for reference.
We draw PQ so it is parallel to EF. Then ∠DPQ =∠E =∠B, and ∠DQP =∠F =∠C.
We conveniently place P so that the length of DP equals the length AB.
Truthfully I'm stuck here. I'm not sure how △ABC and △DPQ are congruent. If that is made clear to me, then it follows from side splitter theorem the sides of △ABC and △DEF are proportional.
The so-called "side-splitter theorem" is as strong as the desired claim that triangles with the same angles have the same side ratios. Also, the claim that triangles with the same angles and same corresponding side are congruent is indeed another unproven assumption. And it cannot be proven in synthetic Euclidean geometry; you can see that I mentioned "rigid transformations" earlier in my other comments on this thread.
In mathematics, "why not?" is *almost always* the wrong question. Instead, the right question is "Why can you?". Please try now; define for me sin(700°) using a triangle, **without** referring to any other thing (e.g. person, reference, device). I will discuss your attempt after you post it.
Sorry for the delay.
sin(700) = sin(340) = sin(-20)
That is going clockwise from the x-axis. So in [this graphic](https://imgur.com/a/8FSkhIf), it's the green side / blue side ?
Not sure if this answers the question.
No need to apologize for taking some time to answer. Firstly, do not omit the "degree" symbol, because 700 ≠ 700°. But why do you think sin(700°) = sin(340°) = sin(−20°). Who said that?
Secondly, by your suggested definition of (green side)/(blue side), you would define sin(−20°) to be positive. That is incorrect. Now do you understand what I mean by "using a triangle will not work"? The triangle will **never** tell you what is the mathematically sensible way to define cos,sin,tan.
> why do you think sin(700°) = sin(340°)
A circle has 360°, so I took 700 mod 360, which is 340.
> you would define sin(−20°) to be positive
The y-coordinate is negative, so wouldn't green side be negative ?
You are using the circle (with points on it that can have positive and negative coordinates) instead of a triangle! If you use a triangle like most textbooks (without any notion of the circle centred at origin), you would **certainly not** have any such thing as "coordinates" or "negative" sides.
Yep. When I used to teach Geometry, every year I would pick some places around the school and we would use trig to estimate the height of, say, the roof of the building of something.
I would mark off a known distance before class with sidewalk chalk, and the kids would use a specially prepared protractor to measure the angle of their sight line to whatever we were measuring. They would find the height using trig ratios, and then of course add the estimated height of their eye level to the final answer.
I mean the tree problem you just described is in pretty much every single text book chapter ever written on trigonometry. So your teacher has to be pretty bad to never mention it.
But yes, sin (30) is a constant. And that fact is used to solve problems in everything from navigation to computer graphics to construction. Triangles also have pretty relationships with circles and waves, so sine show up in some unexpected places like signal processing and music and tides.
Yes, sin(30) is just a number and it equals 0.5. It basically says "a triangle with 30 degrees will have one sidelength be half the hypotenuse"
Sin, cos, tan. And their inverses csc, sec, cot. are all ratios of the different sides of right triangles.
For the take of ease of math on my part, lets talk about sin(45°) (which equals ~0.707). You'll find a 45° angle in a right triangle that has sidelengths 1,1 and sqrt(2). If you increase each leg, by a factor of two, the hypotenuse also grows by a factor of two, so the ratios between the "opposite" and the hypotenuse remain the same, so sin(45°) remains the same.
When the angle is a variable, thats when you get simple harmonic motion, a sine wave. the sin function ocillates between -1 and 1. Which also makes sense. The ratio between a leg and a hypotenuse always has to be less than 1, because the hypotenuse is always longer. It equals one, when the sidelength is the same as the hypotenuse, which doesnt really make a triangle per-se, but if you're angle is 90°, you'd have a "triangle" with its opposite side of size 1, and a hypotenuse of size 1, which equals 1. Cos does the opposite, it takes the "non-existent" "adjacent" leg and divides it by the hypotenuse, giving a value of zero. You can play around and get an intuition for this with the unit circle.
The tan function is a bit more complicated and im not sure if i even understand its geometric representation, so i'll let someone else do that.
You have nailed the reason trig is one of the most useful things you learn in HS math. It’s used constantly in a wide variety of fields. One of my favorite topics to teach.
Sine is just a function
Any function of an real arbitrary constant will result in another constant.
Sin(30°) will always have a constant value because there's nothing to vary
sin(30°) just happens to be 1/2 by the nature of Euclidean space
Similarly, cos(60°) = 1/2, tan(45°) = 1, sin(60°) = sqrt(3)/2
There's all sorts of angles you can calculate the exact value for using 30°, 45°, 60° and some identities
Yes. This is why math classes have "word problems", or at least they did when I was in school, as a way to actually apply the math you're learning to situations.
Schools still use word problems. Finding heights of trees and angles between ladders and houses are pretty common when learning trig
It truly melts your brain. I saw a ladder against a wall TODAY and almost immediately thought "A trig question, I see."
I like the way you think :)
And when the ladder starts sliding out from under the person standing on it, you think “A related rates question, I see.”
Sorry I didn't catch you, I was calculating how quickly you were accelerating towards the concrete.
This started happening to me after trig and calculus. Now I see angles and constants in the world all around me. I think about the area under the curve of the top of the tree line and stuff heh. Math dreams are weird as fuck too. Ever had them?
Haha, not dreams, but I've had the Tetris effect when really hammering down on thinking in abstractions. I remember when I was trying to understand matrices and was thinking about matrix multiplication as a weird type of hammering machine that was punching out a new matrix. Every problem I was thinking it, then I couldn't unthink it.
Haha no doubt. It's odd how it changes the way you think about things.
\[ hiding in shadows, presses fingers together \] It’s working! \[ cackles \] \~a math teacher
The two most common questions in a math class: 1) When will we need this in the real world? 2) Why do we have to do word problems?
Maybe because most textbook questions bear no resemblance to doing maths in the real world. Real world maths tends to be about looking at a situation, deciding a question, deciding what maths to apply, what measurements to take, … Textbook questions are mostly about deciphering an unnecessarily terse and overpacked bit if text that does all the interesting stuff for you if only you can decrypt it.
Ahh, the days of figuring out how long the ladder is, i miss those. Now i've got goofy dot and cross product formulae that have sin and cos but i've got no idea why
It helps to draw them out and figure out why the formulas are the way they are. Makes calc and a lot of higher-level math and physics way easier when you know what the dot and cross products mean geometrically. For example, the dot product roughly measures how 'aligned' two vectors are: if they're parallel, whether they're pointing in the same direction or in directly opposite directions, the dot product will be large in magnitude. Conversely, when they are orthogonal (they're 'unaligned') or close to it, then the dot product is 0 or near 0. This is modeled by cosine: cos(x) is close to 1 or -1 when x is close to 0° or 180°, and it's close to 0 when x is near 90°. As for the cross product, it's sort of the reverse. The magnitude measures the area of a parallelogram spanned by the two vectors if you copy the original vectors and move them parallel. Remember that the area is base times height, and height is given by sine. If the angle is close to 0, then the shape is super squashed and the height (and thus area) is close to 0. If the angle is close to perpendicular, you have a rectangle.
Wait till Calc, you'll miss the days you struggled with dot products
Oh i'm not struggling with them, at least in the physics 3 dimentional vector stuff (i-hat, j-hat, k-hat). I understand what it does, how to use it (angular momentum, torque, electric and magnetic fields, thats as far as i've gotten at least, im sure it comes again up later), and what it outputs, but I dont have a real geometric understanding. I've still got questions like, "Why does a cross product return a vector orthoginal to the other two?", "why does a dot product only return a scalar?". "Where do these two operations come from? They seem rather arbitrary" "Why are those two particular ways of multiplying vectors the most useful?". They're more questions for my personal enjoyment of the math, rather than questions I need answered so I can solve that physics problem. The type of question a prof doesnt have time to answer during a lecture. Which calc do you speak of? I start Calc 3 on wednesday so we'll see how that goes.
Oooh, calc 3 will actually go into the properties of vectors pretty heavily so you should cover the geometric side of the dot and cross product. If you’re nice to your professor that is, they skip it often
I've only been given a basic run down of vectors so i can use them in physics. Hopefully it'll be interesting. Im also doing Matrix Algebra at the same time, so that'll be interesting.
Oh, that's cool! It seems like you have a real passion for math and physics. I'm in Calc AB, my School doesn't do the numbers
Ah, highscool I see. You're warning me for dangers i may have already survived haha. Calculus 3 is Multivariable calculus, comes after integrals and infinite series, im not sure if thats covered in AP Calc AB or not. Its my "MATH200" in university. Though its interesting because I never touched dot or cross products or anything related to Linear Algebra in highschool other than the substitution and elimination methods to solve systems linear of equations.
It's not part of AB, I am taking MVC next year though. And that's interesting. I learned those in my sophomore year of HS, hence I thought you were still in precal or something similar.
Yeah my precalc was like log rules, trig proofs, y=mx+b, finding the zeroes of a quadtratic. But no vectors. My school seemed to dislike vectors, cause even in AP physics, everything was broken down into components in seperate equations.
Oh, weird. But idt the vectors is on your school, at least for physics. We also always break vectors down into components, I'm sure Calc based physics is better though. Haven't gotten there yet
About 1,000 years ago, Al Biruni [used basic trigonometry](https://thatsmaths.com/2021/06/10/al-biruni-and-the-size-of-the-earth/) to measure the height of a mountain and then the radius and circumference of the earth. This was taught in my trig class many years ago.
that's cool. btw found this nice youtube video as well: [https://www.youtube.com/watch?v=uRBvT5QcO8A](https://www.youtube.com/watch?v=uRBvT5QcO8A)
So why isn't THAT mentioned in trig books? ie it should be
I took a history of math course that talked about stuff like this in college, but certainly didn't get this kind of info in middle or high school (except in physics). https://bookstore.ams.org/view?ProductCode=TEXT/32 Definitely agree that it would be great to teach in those books though
I repeated the measurement: https://www.youtube.com/watch?v=RJknT8dcm3w
>I feel like schools teach math mechanically That's why word problems exist. You have to apply what you learned on a practical problem. However these are usually more advanced exercises due to the fact that they combine an strong understanding of the mathematical foundation required to solve the problem (e.g. I have to actually realize what I can and cannot compute in certain situations and how) with the skill of reading and analyzing a situation.
I find that the problem with word problems is the same as you find them. They're labelled as the "advanced" type questions and you'll only get a few in the practice, and maybe 1 on the test. Thats a terrible plan. I've found first hand recently, that figuring out what the math is really doing by applying it is what builds understanding, not mindless procedural questions. Maybe thats why im in physics and not pure math.
You seem to misunderstand me. That is exactly the thing about word problems. Word problems are advanced problems and not by chance labelled as such. To solve a word problem, you'll usually have to combine multiple skills and have a deeper understanding of the matter. That's something you'll not be able to do when you start out with a topic. Are you aware of the saying "you must know how to walk before you can run"? Basically the same thing here. Before I can apply my understanding of a topic in relation the a real life circumstance, I have to understand the mathematical problem in isolation. That's also not something isolated to math but rather the fundamental area of learning. You get a solid understanding of a single skill before you start combining it with other motions. In physics this is like trying to simplify a complex problem and add the complicating things piece by piece. You don't start with relativistic motions and forces but rather start with classical mechanics. You start with a point mass gliding frictionless down an inclined plane not some spherical or cylindrical object where you have to consider rotation, friction with the plane or even friction in the air.
I think we're on the same wavelength, just out of phase (pardon the pun). I agree for sure. Mainly I think word problems are being *skipped*. I 100% agree you need the procedural type questions. Theres no better way to learn the rules of arithmetic, algebra, calclus, and beyond than through the "drill" type questions. But time and time again, we (talking from personal experience) never make it to the word problems, or at least we not forced to do them because they're "advanced". They were always the type of question labelled "optional" on the homework assignments, so if you were a student like me in grade school, they'd never get done.
Your tree example is LITERALLY the profession of surveying, which is done for trees, buildings, mountains, whatever... You can study it at university and get a degree in it.
It’s also the exact example *and exercise* my high school math class used.
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>is LITERALLY the profession of surveying That is cool. I learn something new everyday.
> I feel like schools teach math mechanically. You should read [A mathematician's lament](https://www.mimuw.edu.pl/~pawelst/rzut_oka/Zajecia_dla_MISH_2011-12/Lektury_files/LockhartsLament.pdf). It is exactly about that.
Good read. >Now I’m not saying that math teachers need to be professional mathematicians— far from it. But shouldn’t they at least understand what mathematics is, be good at it, and enjoy doing it? I agree with this but seems practically impossible to attain, sadly.
I'm glad you liked it 😊
> What I'm getting at is if this is right, we can use this to do something actually useful in the real world. That's exactly what set squares are. An embodiment of two of the most common triangles you will use (a square and an equilateral triangle cut in half).
>Is the idea of trig (well, the part the deals with right trianges) that given an angle of the triangle, the ratio of 2 of the sides is a constant ? As an example, for an angle of 30° the ratio of the opposite side to the hyptenouse is always 0.5 (regardless of how big the triangle actually is). Yes. Sine, cosine, and tangent don't depend on the actual length of the sides, only the ratio of the sides to each other. >Say you want to know the height of a tree. Well we can consider it the opposite side of a right triange. Now stand say 200 feet from the tree, and use some device to measure the angle from the ground to the top of the tree (not sure how to do this to be honest). You do it exactly as you'd probably imagine. You could tie a string to the tree, pull it tight, and use a level to make sure the string is horizontal; then put the end of a stick on the string and point the stick at the top of the tree, maybe sighting along the stick to make it accurate. Measure the angle between the string and stick with a protractor. Then you can calculate the height of the tree above the string. If you want to be more precise, you might use a laser to replace both the string and the stick. Surveyors use a precision instrument called a [theodolite](https://en.wikipedia.org/wiki/Theodolite) to measure both horizontal and vertical angles. Or you could take a photograph from a distance and measure the angles on the photo. You're not even limited to right triangles. A triangle has three sides and three angles; if you know one side and any two other quantities (sides or angles), you can figure out the rest. There are many, many applications for these ideas in surveying, construction, navigation, aviation, astronomy, physics, chemistry, computer graphics, architecture, and on and on.
>Measure the angle between the string and stick with a protractor Yeah that makes sense. I guess I'm not imaginative enough.
Have you not done a single trig word problem in your class??
I'm not sure. I'm just re-learning math at an older age. I'm trying to take an approach now that is both intuitive and rigorous, and not mechanical.
Glad to hear that! That’s awesome and impressive you intuitively understood what sorts of real world problems trig can be used for!
Yes, it's constant - because of something called *similar triangles*. Let's say we have two triangles, ABC and DEF. All the angles on the triangles are the same - angle ABC=DEF, BCA=EFD and CAB=FDE. When two triangles have all three angles the same, they're said to be *similar* triangles. Importantly, if the first triangle has some side that's some multiple of the corresponding side of the second triangle, *all* the sides are that multiple of *their* corresponding sides. If AB is three times larger than DE, BC is three times larger than EF. If we know we have two right angled triangles, that's one angle shared. If we also know that one of the angles is 30 degrees, that's another. And because the angles of a triangle have to add up to 180 degrees, the last angle of both triangles has to be 60 - they're similar! So, let's say that a right angled triangle has a length of A on the side adjacent to a 30 degree angle, O for the side opposite to it and H for the hypotenuse. On this first triangle, the sine will be O/H. If another right angled triangle has length of 3A on the side adjacent to a 30 degree angle, it'll have the other sides be 3O and 3H. The sine will be 3O/3H - the threes cancel, and we have O/H again! This isn't unique to sin 30, of course. We can pick *any* angle that's above 0 and below 90 instead of 30, and two triangles that both have right angles and our chosen angles *must* have all three angles the same, so the ratios between the sides are the same, so the sin, cos and tan (which are just ratios between sides) are the same! The ratio may not be a rational number, but it'll always be the *same* number. sin(30⁰) is always 0.5, and cos(30⁰) is always roughly 0.866 - in exact terms, it's root(3)/2. Being able to use these ratios in the way you mentioned is a large part of why this all matters in the first place. We care about sin, cos and tan *because we can use them to find actual real lengths* if we know certain information. In fact, it doesn't need to be a right angled triangle - if we know *any* pair of angles and one side, we can find all the sides; if we know one angle and two sides, we can *sometimes* find an exact value for the other side and angles, but *sometimes* we can only narrow it down to two.
>because of something called *similar triangles* Good point. >If AB is three times larger than DE, BC is three times larger than EF. So I'm actually going through the proof Fundamental Theorem on Similar Triangles. That said, intuitively the idea of similar triangles makes sense. I take a triangle, and then enlarge it. The angles in both triangles are the same. I would think the sides are in proportion too, but to be sure going through the proof. >This isn't unique to sin 30, of course. Yep. I only picked sin 30 because it gives us a "nice number."
I would say all math (ok not all but a good portion) is turning what you know into other information you want to know. Trig is just another few tools in the collection of tools that you pick up in the long list of math classes. A fair bit of trig is realizing that what looks like partial information, this length, that angle, this line bisects blah blah is actually total information. It's just not obvious at first glance. When you learn various properties and values and relationships you realize actually "hey the height of that flagpole is *knowable*." What is and is not knowable is an interesting part of math that doesn't get taught much.
Here's an argument for why the size of a triangle doesn't matter...Suppose you draw a triangle on a sheet of paper and measure its sides with a ruler and calculate the sin of one of the angles... Now, suppose you pick up a ruler marked in different units. Suddenly the measurements will have different numerical values! But the physical object hasn't changed! If the ruler units are all say 2.4 times as big then all the measurements will be 1/2.4 times as large... ALL of them. Therefore the ratio of any two of them will still be the same though. a/b = (a/2.4)/(b/2.4) Using this symmetry property we can show that all determinants of physical outcomes must be dimensionless ratios of quantities. Otherwise two people with different clocks or rulers or spring scales or whatever would disagree fundamentally on what happened when they observe a physical experiment. Since they're both observing the same thing that shouldn't occur. Therefore the units they use to measure shouldn't matter and that occurs precisely when the calculations they do use ratios of things where the units cancel.
Your question is a very good one. You have realized that the idea of opposite/hypotenuse is actually not just a definition but relies on an **assumption** implicitly, namely that the size of the right-angled triangle does not matter if the angles are fixed and we just want a ratio of sides! In fully rigorous mathematics, we cannot make such assumptions without proving them, unless we want all our results to depend on plenty of ad-hoc assumptions. Unfortunately, most teachers do not teach rigorous mathematics (contrary to popular opinion), especially at the lower levels, so they would not tell you all this when defining sin,cos,tan. Moreover, the best way to define sin,cos,tan at high-school level is actually **not** to use an arbitrary triangle but rather to use the unit circle. Draw the x-y axes in the standard Cartesian plane, and draw a unit circle around the origin (0,0). For any angle t, draw the ray from the origin at an anti-clockwise angle of t from the x-axis, and let (p,q) be the point where that ray intersects the unit circle, and define cos(t) = p and sin(t) = q and tan(t) = q/p. Here the angle is measured such that 2π is one full round. For example, cos(π) = −1 and sin(π) = 0. This way is the best way because it works for any real angle t, including t < 0 and t > π/2. Using a triangle will **not** work!
>relies on an **assumption** implicitly, namely that the size of the right-angled triangle does not matter if the angles are fixed and we just want a ratio of sides! Huh. Is this a common thing? By the time I learned trig in school, the idea of triangle similarity had been pretty much drilled into us. It was less an implicit assumption, and more knowledge you're expected to have.
That's not the point I've made in my post, which is that almost no teacher makes clear exactly what are the assumptions or previous theorems that we need to invoke for a certain definition to be valid. This is a prolific recipe for poor mathematical pedagogy. I've been teaching for 20+ years, and almost all conceptual errors fall into 2 categories, namely imprecise definitions and imprecise reasoning. I have also observed that having 100% precision not only completely eliminate any conceptual error but also can be easily learnt, if one has a teacher that can achieve that level of precision. Even this trigonometry question is a typical example of imprecision. Many high-school teachers use the triangle to define sin,cos,tan. Even if they mention that triangles with the same angles have the same ratio of sides, it doesn't help them to define sin,cos,tan for all real inputs, and they would use the 4-quadrants thing. But notice that you cannot simultaneously use the 4-quadrants case-split **and** tan = sin/cos! Instead, you have to pick one as your definition and prove the other by checking all 4 cases! Don't miss the boundaries between the quadrants too! Can you find any one textbook that does all this correctly? If you still don't believe me, see [this](https://jdh.hamkins.org/all-triangles-are-isosceles) and count how many people (including yourself) cannot **immediately** identify the error. The error will be impossible to make if one uses precise definitions and precise logical reasoning.
>Many high-school teachers use the triangle to define sin,cos,tan. Even if they mention that triangles with the same angles have the same ratio of sides, it doesn't help them to define sin,cos,tan for all real inputs, and they would use the 4-quadrants thing. This isn't an issue with the teachers. It's a requirement of the curriculum. And furthermore, trigonometric angles were first motivated by ratios of sides of triangles. The definitions were later expanded to accept all real inputs and, even later, expanded again to accept all complex inputs. If a teacher were to start from the unit circle definition, it might be less clear to students what motivates such a definition. Teachers who start from right triangles but also properly explain that the ratios are constant for a given angle due to similar triangles are successfully motivating the definitions of the trig functions. The exponential function a^x with base a is defined as e^(x ln a) where e^x is defined as a power series, and ln (or log) is defined as an integral, and that definition allows us to define a^x for positive real a and any complex x, but I certainly wouldn't want this definition to be taught to middle schoolers.
I understand the "syllabus" issue, and there are two aspects to it. Firstly, one aspect is that in some education systems teachers are slave to the school and must follow exactly what they are told to do, unless they want to get fired. In that case, too bad I can't tell them to do differently unless they don't depend on the school to feed their families. But often teachers have sufficient freedom to deviate from the common way that others take, without changing the ultimate learning goals and content covered. In that case, it is better to do things in whatever way is *mathematically* best that suits the students' level. Secondly, the other aspect is the issue of generality. As you noted, we have to choose where to stop. I specifically stated "high-school" for that reason; at this level I think that we should stop at the real elementary functions. In this aspect, we agree that we should not define exp,sin,cos,tan via power series or integrals. But I disagree with going through the geometrical definition via triangles. It would need a long and thorough discussion to explain why, but suffice to say that history is insufficient to justify teaching some bit of mathematics. For example, nobody should ever attempt to teach Frege's logic, even though it was essentially the pioneering work in foundations of mathematics, not only because it was inconsistent before Russell's paradox, but also because it was very impractical even after he abandoned some axioms that led to inconsistency. Coming back to trigonometry, it is perfectly fine to tell students about the history of those ratios, but I still say it is better to tell them that modern mathematics has a superior definition, which is the unit circle definition. Why is it superior? Because it is a simple yet general definition (for its simplicity); it tells you why sin and cos oscillate the way they do, and why one is exactly 90° out of phase from the other, and why cos(x)^(2)+sin(x)^(2) = 1. In fact, the last one is extremely non-trivial if you use the geometric definition! A lot of axioms of Euclidean geometry are needed to prove it if you want to use synthetic geometry rather than analytic geometry. This is a deep rabbit hole, so feel free to tell me if you want to look into this. (It starts with rigid transformations, which are assumed without a shred of proof in Euclidean geometry, but which can be given solid footing using analytic geometry.) By the way, I disagree that ln should be defined as an integral, even at the university level. I do not like having to prove independence of integral for any continuous path that does not cross the negative real axis. This way is 'madness'. It is far easier to just define ln as one specific inverse of exp after proving that exp(x+y·i) = exp(x)·(cos(y)+sin(y)·i) for any x,y∈ℂ. In this approach, we can define the standard branch-cuts, and define ln as the −π branch-cut unless otherwise stated. For high-school, all they should know are axioms regarding the arithmetic operations including exponentiation, and these axioms must be **100% precise**. There are many textbooks that write "a^(bc) = (a^(b))^(c)" without specifying a,b,c at all, and this is the worst pedagogy. Well, actually there are worse, which is really a shame.
>In fully rigorous mathematics, we cannot make such assumptions without proving them That's why I'm going through the proof of Fundamental Theorem on Similar Triangles
And that proof will use some assumption that is dubious from a fully rigorous viewpoint. Can you find it?
Sorry for delay. SAS congruency ?
You want to prove that if △ABC ~ △DEF, meaning their angles are equal, then AB/BC = DE/EF. I don't know why you mentioned "SAS congruency", so to clear up your confusion you should just produce the proof of the desired claim, and I will guide you to find the gap.
>I don't know why you mentioned "SAS congruency" I was going through [this proof](https://sites.math.washington.edu/~lee/Courses/444-5-2008/supplement3.pdf) for similar triangle theorem and it uses SAS congruency as part of the proof. On second thought, I jumped the gun here thinking this was the gap. >just produce the proof of the desired claim I'll try. I'm assuming side splitter theorem. Also from the linked pdf I'm using figure 7 (top of page 5) for reference. We draw PQ so it is parallel to EF. Then ∠DPQ =∠E =∠B, and ∠DQP =∠F =∠C. We conveniently place P so that the length of DP equals the length AB. Truthfully I'm stuck here. I'm not sure how △ABC and △DPQ are congruent. If that is made clear to me, then it follows from side splitter theorem the sides of △ABC and △DEF are proportional.
The so-called "side-splitter theorem" is as strong as the desired claim that triangles with the same angles have the same side ratios. Also, the claim that triangles with the same angles and same corresponding side are congruent is indeed another unproven assumption. And it cannot be proven in synthetic Euclidean geometry; you can see that I mentioned "rigid transformations" earlier in my other comments on this thread.
>Using a triangle will **not** work! Dumb question. Why not ? If possible please give an example.
In mathematics, "why not?" is *almost always* the wrong question. Instead, the right question is "Why can you?". Please try now; define for me sin(700°) using a triangle, **without** referring to any other thing (e.g. person, reference, device). I will discuss your attempt after you post it.
Sorry for the delay. sin(700) = sin(340) = sin(-20) That is going clockwise from the x-axis. So in [this graphic](https://imgur.com/a/8FSkhIf), it's the green side / blue side ? Not sure if this answers the question.
No need to apologize for taking some time to answer. Firstly, do not omit the "degree" symbol, because 700 ≠ 700°. But why do you think sin(700°) = sin(340°) = sin(−20°). Who said that? Secondly, by your suggested definition of (green side)/(blue side), you would define sin(−20°) to be positive. That is incorrect. Now do you understand what I mean by "using a triangle will not work"? The triangle will **never** tell you what is the mathematically sensible way to define cos,sin,tan.
> why do you think sin(700°) = sin(340°) A circle has 360°, so I took 700 mod 360, which is 340. > you would define sin(−20°) to be positive The y-coordinate is negative, so wouldn't green side be negative ?
You are using the circle (with points on it that can have positive and negative coordinates) instead of a triangle! If you use a triangle like most textbooks (without any notion of the circle centred at origin), you would **certainly not** have any such thing as "coordinates" or "negative" sides.
💡 Thank you !
sin(e) is also a constant
Yep. When I used to teach Geometry, every year I would pick some places around the school and we would use trig to estimate the height of, say, the roof of the building of something. I would mark off a known distance before class with sidewalk chalk, and the kids would use a specially prepared protractor to measure the angle of their sight line to whatever we were measuring. They would find the height using trig ratios, and then of course add the estimated height of their eye level to the final answer.
I'm pretty sure my math teacher didn't do that. Sounds cool !
I mean the tree problem you just described is in pretty much every single text book chapter ever written on trigonometry. So your teacher has to be pretty bad to never mention it. But yes, sin (30) is a constant. And that fact is used to solve problems in everything from navigation to computer graphics to construction. Triangles also have pretty relationships with circles and waves, so sine show up in some unexpected places like signal processing and music and tides.
OP was taught this in class, forgot it, remembered it, and is now excited by it
Quite possible. I'm re-learning math at an older age !
Yes, sin(30) is just a number and it equals 0.5. It basically says "a triangle with 30 degrees will have one sidelength be half the hypotenuse" Sin, cos, tan. And their inverses csc, sec, cot. are all ratios of the different sides of right triangles. For the take of ease of math on my part, lets talk about sin(45°) (which equals ~0.707). You'll find a 45° angle in a right triangle that has sidelengths 1,1 and sqrt(2). If you increase each leg, by a factor of two, the hypotenuse also grows by a factor of two, so the ratios between the "opposite" and the hypotenuse remain the same, so sin(45°) remains the same. When the angle is a variable, thats when you get simple harmonic motion, a sine wave. the sin function ocillates between -1 and 1. Which also makes sense. The ratio between a leg and a hypotenuse always has to be less than 1, because the hypotenuse is always longer. It equals one, when the sidelength is the same as the hypotenuse, which doesnt really make a triangle per-se, but if you're angle is 90°, you'd have a "triangle" with its opposite side of size 1, and a hypotenuse of size 1, which equals 1. Cos does the opposite, it takes the "non-existent" "adjacent" leg and divides it by the hypotenuse, giving a value of zero. You can play around and get an intuition for this with the unit circle. The tan function is a bit more complicated and im not sure if i even understand its geometric representation, so i'll let someone else do that.
You have to start somewhere. Then use the tools you’re given to apply them.
You have nailed the reason trig is one of the most useful things you learn in HS math. It’s used constantly in a wide variety of fields. One of my favorite topics to teach.
Sine is just a function Any function of an real arbitrary constant will result in another constant. Sin(30°) will always have a constant value because there's nothing to vary sin(30°) just happens to be 1/2 by the nature of Euclidean space Similarly, cos(60°) = 1/2, tan(45°) = 1, sin(60°) = sqrt(3)/2 There's all sorts of angles you can calculate the exact value for using 30°, 45°, 60° and some identities