I’ve read the definitions of this trigonometric functions, but don’t understand their real-world import. Can someone give a few examples/applications?
Also this: I’ve seen movies now and then, in which a Navy technician is observing a screen (or possibly an oscilloscope) and says something like this to his chief, “I’m getting a classic sine wave over here.”
If you’re ever trying to do any real-world geometry things that concern angles and distances, you can’t escape needing to know the trig functions. Everything can be broken down into triangles, and these functions describe the triangles. Combine them with the law of similar triangles, and the Pythagorean Theorem, and you’re pretty well-equipped.
Oh, about the “sine wave” - if you graph the height of your friend on a Ferris Wheel that’s moving at a steady speed, so that time is the x-axis, and his heigh is the y-axis, you can imagine that he goes from the minimum height to the maximum height along a smooth curve, that’s rounded-off on the top and bottom. This is really just a graph of the sine function, and is called a sine wave. They’re ubiquitous in nature. The motion of a weight on a spring, waves on water, sound waves, electrical waves, all are closely underpinned by an understanding of sine waves.
I know the answer to the first half of the question, but I don’t know how to best explain it. I might post again in a few hours when I’ve figured it out, unless someone else gives a better explanation.
As for the second half, a sine wave is what you get when you plot a graph y = sinx . You’ve probably seen it before; it looks like this. I don’t really know what a Navy technician would be talking about when he says that though.
A sine wave is what you get when you plot the sine function on a graph. The simplest one is y = sin(x). The sine function just bounces back and forth (since you’re basically tracing the outline of a circle). So the graph looks like a very simple wave.
Sine waves are very easy to produce electronically, and can be viewed on an oscilloscope, for example.
BTW, I meant to add: If you look at the circle, the length of the red line is the value of the sine function for that value of x. You see how it’s just the leg of a triangle with the radius as the hypoteneuse? That’s the basic idea of trigonometry.
The cosine function is the same thing except it measures the other leg of the triangle. (The one adjacent to the angle, where the x axis intersects the circle on that picture.)
Or, more generally, any cyclic process or motion can be described using trig functions, which is why trig functions abound in physics and engineering. Even more general than that, as ultrafilter mentions, using Fourier series it is possible to describe any periodic function using nothing more than basic trig functions. It’s something that, like calculus, is very important to describe the world and useful in its application, but an intimate knowledge isn’t necessary to most people in their day-to-day lives. As an example, without trig most electronics (circuits) would be pretty hopeless, so even though you may not need to know trig to listen to your mp3 player or use your computer it’s partially thanks to trig that you have them.
simple example from the construction business:
Try to get a permit to build a tall building opposite a small private airport. Airplanes take off at a certain sloping angle. The cosine of that angle, times the distance between the building and the runway tells you how tall the building can be without the plane crashing into it.
Geographers and others who do remotes sensing stuff (sattelite images and the like) need to use trig functions to figure out how to correct for the sun angle at the time an image is taken – so they can, say, compare two images and so track things like land use change or crop growth over time.
The g forces that the occupants of an airplane are subjected to in a coordinated turn while maintaining a steady altitude are the inverse of the cosine of the angle of bank.
This may sound a bit obscure, but it’s important because if you bank too steeply while performing such a turn in an airplane one of two things are likely to happen: 1) the airplane is no longer able to generate sufficient lift to remain at the “steady altitude” and you will start descending in a precipitous manner, thereby causing great mental distress to the occupants of said airplane, not to mention the potential for hearing damage from all that screaming or 2) you will exceed the engineering limits of the machine and break something - which happened not too long ago at a local airport I fly from, when someone broke (made one piece into two) the main wing spar of one of the airplanes doing pretty much just that, too much bank angle (fortunately, no one got hurt although they are having to replace many structural compents of that airplane).
Is that relevant enough to reality to satisfy the OP?
Actually, pilots do not sit in the cockpit computing these numbers - we tend to go by the rule of thumb that, in most cases, exceeding a 60 degree bank or 2 g’s in a turn is Not A Prudent Idea. Very few passengers are ever likely to see more than 45 degrees, and if I recall airlines tend to keep it to 30 degrees or less (barring life or death emergencies). In my experience, pilots who do exceed 60 degrees of bank regularly and delibrately tend not to carry passengers, are supposed to wear parachutes, and do pay attention to both their bank angle and their g meters even if they are not explicitly computing trigonomic functions while in flight.
They’re also important when sailing or flying: if you’re sailing in one direction and the current is moving in another then you’re not going to end up where you want, so you need to adjust your course. To do that involves mathematics. Over a short distance, you can approximate to a triangle.
Or consider a submarine firing a torpedo: you don’t fire at where the target currently is, you fire at where it’s going to be. Basic trig again.
In ordinary everyday affairs they aren’t of any consequence to most people. However, they are of great importance in scientific and technical work.
For example, in electrical engineering, the output of a generator that consists of a rotor rotating in a uniform magnetic field is a sinusoidal wave. That is, the amplitude of the volatage at any instant equals the sine, or cosine (they are the same curve displaced from each other in time) of an angle.
Also any arbitrary, repetitive mathematical function can be represented by a series of sinusoids of various amplitudes by the process of Fourier series expansion. Any waveform whatever can likewise be so represented by a related technique called the Fourier transform. This allows engineers to compute the bandwidth required to pass an arbitrary electrical signal with little distortion and that’s important in communication systems. And, unless I misunderstood what I read about Quantum Mechanics, the Fourier transform is of importance there.
Sinusoid waves underlaie the analysis of music, speech, the frequency and shape of waves on a beach and endless other natural phenomena.
Measure the angle of your sight lines to the top and bottom, then pace off the distance to the tree. Trig functions allow you to measure angles and distances indirectly by using the properties of triangles. With precision instruments for measuring the angles, you can be as accurate as you like.
Very important. Position and momentum, in quantum mechanics, are related via a Fourier transform (and a factor of hbar, but that’s not important here). This is the origin of the Uncertainty Principle: Fourier transforms turn broad distributions into narrow ones, and vice versa.
I use them at least weekly, but my work’s proprietary.
However, for fun, I survey historical ruins. When I try to match CAD drawings to GPS measurements I include rotating the drawing in my statistical model, and adjust the angle and offsets of the drawing to make drawing x and y match longitude and latitude. There are sines and cosines throughout these equations.
And I’m monitoring my blood pressure, because it measured high recently - I’m going to model the daily fluctuation in it, among other things, because I’m interested. So I’ll fit sine(time_of_day) and cosine(time_of_day) as two “factors” (really, offsets, not multiplicands, though in regression analysis for some reason things that add together to build a model are called “factors”).