Something I have wondered since high school: why does trigonometry only use, or mainly concentrate on IAE, right triangles? There are a lot more triangles, than right, I should think. Actually, I think the vast majority are not right. So why do the trigonometric functions only deal with right triangles? And does have to do with it’s used in math and science have anything to do with it?
And then there’s calculus (which I had in college, for the first time, some time back). It is based on the function. I am sorry to tell you. But there are more graphs than just functions. Oh, sure, our HS physics teacher said time never goes backwards (which I theorize could be partly the reason why). But what if you wanted to study something going back in time? Or what if the phenomenon you study simply is not a function at all? What do you do then? And again I ask, does this have some relation to math and science, and how it fundamentally works. Or is it, are they both, just a convention?
Right triangles are a good place to start, because the Pythagorean theorem is simple and accessible. The Law of Cosines is a generalization of this theorem to non-right triangles.
Trigonometry focusses on right triangles because that’s how the trigonometric functions are defined. These functions can, geometrically, all be regarded as ratios between the lenghts of various sides in a right triangle. As simple as this sounds, these functions are tremendously useful for a range of applications in mathematics, physics, geography and other fields.
You can, of course, also study non-right triangles. I would regard this as geometry, however, not as trigonometry.
The trigonometric functions show up in a whole bunch of places that aren’t specifically related to any kind of triangle (e.g. simple harmonic motion, Fourier series, wavefunctions of unbound particles). The role of right triangles in early trigonometric pedagogy is to introduce these widely useful functions in an intuitive way.
One could make a point to talk a lot about non-right triangles in introductory trig, but that adds a lot of complexity to the presentation, without adding any broadly-applicable concepts you don’t get out of right triangles.
Really the same thing here. Studying functions leads to a bunch of widely applicable and useful tools and concepts. Studying various varieties of non function does not.
Besides that, fairly often it is easy to use functions to study non-function things. For example, a graph that doesn’t pass the vertical line test, so it is not manifestly a function from R → R, can still be represented by a function from R → sets of R, or a parametric curve from (arbitrary parameter) R → R^2, both of which are standard functions.
The trigonometric functions are defined based on ratios in right triangles, but since any triangle can be split into two right triangles this gives rise to formulas for any triangle. The law of cosines for instance “extends” Pythagoras to any triangle where you know three out of the six sides and angles, either two sides and one angle or three sides. If you know two angles (so all) and one side instead, you have to use the Law of sines instead. If you only know three angles and no sides you don’t have a uniquely defined triangle.
High school calculus only works for functions, but many graphs that aren’t a function of a variable on the x-axis (which is what high school math usually sticks with) can be defined as a function of a different parameter. To take just one example: A circle in the x-y-coordinate system can’t be described as a single function of either x or y, but you can do calculus by splitting it into two functions, or you can define it as a function in polar coordinates instead.
Basically the answer to your Q does this have some relation to math and science, and how it fundamentally works is, No, because your premises are false. The limitations you perceive are the restrictions of what is taught in highschool, not what exists in math.
Maybe right triangles come into it in the same sense the usual x and y coordinates are orthogonal. Like everybody says, that is the definition of those functions and also of the related hyperbolic trigonometric functions. But it does not follow that trigonometry or trigonometric functions only apply to right triangles any more than the idea of Cartesian coordinates does. Also, as an exercise consider trigonometry on a sphere if you are not used to it.
In calculus or differential geometry you could have some coordinate t representing time in some scientific problem, but more generally coordinates are pretty arbitrary and there is no reason to worry about anything going forward or backwards in or being a function of time. You can study trigonometry and geometry without invoking any physics.
That’s one way they’re defined, and often how they’re introduced; but it’s not the only way. Trigonometry is often introduced as the study of right triangles, but it goes far beyond this.
Got an example? I suspect that any phenomenon that can be described mathematically can be described in terms of functions, somehow.
For example, a curve in the plane, even if it is not the graph of some function y = f(x), could be described with parametric equations where x and y are both functions of some parameter t.
Sure, you can define the functions in a different manner too. But as you say, side ratios in right triangles is the usual way maths education introduces the topic to students. To my knowledge this is also the origin of the development of these functions in the history of mathematics. Because of this, and because triangles are the very thing the OP asks about, I think it’s fair to take this angle (pun intended) on the topic.
Right triangles are much simpler than random triangles. If I know just a little bit about the triangle, I can figure out the rest easily and reliably, and that is exactly what the trigonometry functions are for. But if it isn’t a right triangle, then I’m going to need a lot more information about it to do anything useful.
Plus: All those other triangles can be easily converted to a pair of right triangles, simply by drawing a perpendicular line from the longest side to the opposite corner. Now that you have two right triangles, most of your calculations are lot easier.
So one answer for the OP is: We don’t have formulas for non-right triangles, because we don’t need them.
For example, a right triangle with a 42° angle will be similar to any other right triangle with a 42° angle, so their sides will be in the same ratio. And the six trigonometric functions denote the six possible ratios of one side to another.
But, as noted in this thread, there’s way more to trigonometry to just triangles. Unless I missed it, no triangles appear in this video:
I think what you mean is that trigonometry is useful for figuring out circles and curves and lots of stuff other than triangles, which is very true. But you can’t understand those circles and curves and stuff until you break them down into their component triangles.
Any time one speaks about “hypotenuse” (like in the phrase “ratio, opposite and hypotenuse” at about 3:15), they’re implicitly talking about triangles. And then the triangles actually appeared on screen from 4:58 to 5:10.
Others have adequately covered the fact that trigonometry extends to much more than right-angle triangles. I’d like to offer an alternate approach to understanding the trig functions, this wasn’t the first way they were presented in my education, but is the way that really clicked for me.
Instead of thinking about the relationship between the lengths of sides of a right angle triangle, instead consider a circle with radius of 1 unit, centered on the origin. The coordinates of any point on the circle thus satisfy the equation x^2 + y^2 = 1. If we define an angle A where A = 0 lies in the direction of the positive x-axis, A = 90 degrees lies in the direction of the positive y-axis, then the points on the circle are also defined by x = cos(A) and y = sin(A) and we immediately have the identity (cos(A))^2 + (sin(A))^2 = 1. Most of the rest of the fundamental trig identities flow easily from this, and it provides a nice introduction to the concept of transforming between coordinate systems, x and y to r and theta, or Cartesian to Polar in this case.
Yep, the “unit circle” definition you described is usually taught after right angle trig (think Soh Cah TOA, finding sides and angles, bearings, angles of elevation and depression) is introduced.
That way kids can see what you can do with trig first and then learn the foundations afterwards if they need it for further topics.
As it happens, this is what I’m teaching right now. And so I feel obligated to nitpick: Two sides and one angle is only a case for the Law of Cosines if the angle happens to be in between the two sides (SAS). If, instead, it’s one of the other angles (SSA), then you can’t use the Law of Cosines, but you can use the Law of Sines (with the caveat that in some cases, there may be zero or two real solutions).
Thank you, moes_lotion. I had been locked into seeing right triangles everywhere on the coordinate grid, but you’ve shown me that it is an arbitrary matter of perspective. Replace “hypotenuse” with “radius” and it’s a whole new world.
I’m going to have to spend some time digesting this. Thanks again.
In the case with zero real solutions (for instance, SSA = 5, 10, 45º), there are solutions, but they’re complex (because that’s what you get when you take the inverse sine of a number greater than 1).
Commence discussion over what, if anything, a triangle with complex angles is, means, or is for.
You don’t need functions to do Calculus. For example, I can just declare a relationship between two variables:
x^2 + y^2 = 1
Perturb the point a little:
(x+dx)^2 + (y+dy)^2 = 1
Do a little algebra with the expressions and you get the relationship:
dy/dx = -x/y
That’s just a declaration that the slope of a tiny segment along the curve (a circle in this case) is -x/y. You could turn it into a function by solving for y in terms of x, but then you’d have to pick one of the two halves of the circle, since traditional functions can only have one value.
Functions are used because they are a nice abstraction to get started with, but you don’t have to do Calculus that way.
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