What are the real world applications of sine waves?

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Discussion Overview

The discussion revolves around the understanding and applications of sine waves, particularly in relation to trigonometry and their real-world implications. Participants explore the mathematical foundations of sine, the process of deriving angles from sine values, and the broader significance of sine waves in various fields such as physics and engineering.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning
  • Experimental/applied

Main Points Raised

  • Some participants express a desire to understand the process of deriving angles from sine values, questioning the steps involved beyond using a calculator.
  • Others suggest that understanding the unit circle and the relationship between angles and coordinates is crucial for grasping sine and cosine functions.
  • A participant mentions the importance of visualizing triangles in relation to the unit circle to comprehend sine and cosine better.
  • One participant highlights the connection between trigonometry and periodic motion, noting that sine waves model various phenomena such as music, communication signals, and biological rhythms.
  • Another participant reflects on their initial confusion regarding the calculator's function for sine, later realizing the underlying formulas and finding beauty in trigonometry.
  • There is a mention of the classic solutions to differential equations that describe wave phenomena, emphasizing the ubiquity of sine waves in nature and technology.

Areas of Agreement / Disagreement

Participants generally agree on the significance of understanding the unit circle and the relationship between angles and sine values. However, there remains some uncertainty regarding the best methods to achieve a comprehensive understanding of these concepts, with differing opinions on the necessity of foundational knowledge in trigonometry before applying formulas.

Contextual Notes

Some participants note that their understanding of sine and its applications evolved through exploration and discussion, indicating that initial assumptions and methods of learning may vary significantly among individuals.

Who May Find This Useful

This discussion may be useful for students and individuals interested in deepening their understanding of trigonometry, sine waves, and their applications in real-world contexts, particularly in physics and engineering.

Nano-Passion
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I know what sine is but I don't know what it really means!

How do you get an angle from sin 1/2? Where is the process in between? Plugging it in the calculator doesn't quench my thirst for knowledge and mastery of it.

I don't want to just know it, I want to completely understand it... :bugeye:
 
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No sorry, I want to know how can you what is the process of going from a ratio of two sides to an angle? I feel like something is missing in between.
 
yes, the angle is between the two sides.
 
Nano-Passion said:
...

Its okay, I found the answer in another thread. - https://www.physicsforums.com/showpost.php?p=3250322&postcount=4

You are getting too much ahead now, it's crucial to understand trig first without using that formula, unless you already understand *why* that formula is true from Calculus. So I suppose that answer doesn't really help you to "completely understand"...

The link Caramon provided on unit circle is a good way to understand cos and sin, make sure you try to understand this thoroughly. The basic idea is that cos(A) and sin(A) is just the (x,y) coordinate of the point on unit circle which one obtained by moving by angle A from the x-axis counterclockwise. For example, if you are move 45 degree on the circle counterclockwise from the x-axis, then geometry tells you that the coordinate (x,y) of the point satisfies x=y. Since radius of unit circle is 1, you can solve to get x=y=1/sqrt2. That is, cos(A)=sin(A)=1/sqrt2.
 
Nano-Passion said:
I know what sine is but I don't know what it really means!

How do you get an angle from sin 1/2? Where is the process in between? Plugging it in the calculator doesn't quench my thirst for knowledge and mastery of it.

I don't want to just know it, I want to completely understand it... :bugeye:

Draw the triangle. Place it on a unit circle. Pythagorean Theorem!
 
Really the old "Soh Cah Toa" saying and a unit circle if all you need to "understand" what the functions are actually representing. Most trig problems from a trig course use angles that can be resolved on the unit circle for a reason, after working through the entire course you should be able to just about resolve any of those major angles in your head without needing a calculator (or the unit circle). I suggest you take a look at the Khan Academy's videos on Trig functions to better explain the idea.

http://www.khanacademy.org/#trigonometry

I think trigonometry is wonderfully beautiful, albeit a bit difficult to grasp at first.
 
i always found it somewhat mysterious that something as "pointy" as triangles, which are made entirely from straight lines, are so intimately bound up with circles, which hae 'nary a straight line in sight.

have you ever seen a radar scope? there is a line that sweeps out in a circle from some central point. that line forms an angle, with it's "starting place". this angle increases from 0 degrees (at the very start) up to 360 degrees (as it comes "full circle").

now, the line (ok, technically it's a ray, since it only goes on forever in one direction), intersects the circle of radius r (meaning, of course, "whatever" in math-speak) at some point, (x,y). so when the angle of the ray is θ between its current position and its "home position", we call x/r cosine of θ, and y/r sine of θ. to simplify things, it helps to use a circle of radius 1, so as to make the arithmetic easier. then x = cos(θ), y = sin(θ).

so how do we get a triangle out of all this? we use the 3 points (0,0), (x,0) and (x,y). that gives us a triangle which depends only on the angle θ (which is the angle between our ray, and the x-axis).
 
  • #10
[URL]http://www.mathnstuff.com/math/spoken/here/2class/330/gif/sincosa.gif[/URL]
 
Last edited by a moderator:
  • #11
The real glory of Trigonometry is that it is one of the classic solutions to the Differential Equations which describe all periodic motion. That would be all wave phenomena. Sines and Cosines and their combination perfectly model music, and all the signals for TV, Radio, Radar, Satellite communication, GPS, Cell phones and the acoustics of speech and singing ... and much more.
Sine waves [the sum of different frequencies] describe the beating of your heart, blinking of your eyes, your breathing, the rotation of the Earth around the sun, etc, etc

The sine wave connection to the triangle is best seen in this previous post #10 gif.
You can see that as the point rotates at some frequency, the position in one plane will trace out the classic sine wave. And this is true if you look along any axis. You can see it bobbing up and down. The waveform generated will be the sine wave vs time if the rotational speed is constant.

Similar ubiquity is found with the exponential e to the ax, but that is another story.
 
  • #12
Sorry for the late reply. I should say that I figured it out by the way, the reason that it was driving me insane is that I was thinking about it too hard. It was a little different from what I am used to in algebra. I didn't understand how you plug in sin 1/2 in the calculator and out comes a degree measurement. But I understood that a calculator uses a long formula that I wasn't aware of.

After I was aware of it, everything clicked. I got a 100 on my test and I see the beauty of trig now =D

paulfr said:
The real glory of Trigonometry is that it is one of the classic solutions to the Differential Equations which describe all periodic motion. That would be all wave phenomena. Sines and Cosines and their combination perfectly model music, and all the signals for TV, Radio, Radar, Satellite communication, GPS, Cell phones and the acoustics of speech and singing ... and much more.
Sine waves [the sum of different frequencies] describe the beating of your heart, blinking of your eyes, your breathing, the rotation of the Earth around the sun, etc, etc

The sine wave connection to the triangle is best seen in this previous post #10 gif.
You can see that as the point rotates at some frequency, the position in one plane will trace out the classic sine wave. And this is true if you look along any axis. You can see it bobbing up and down. The waveform generated will be the sine wave vs time if the rotational speed is constant.

Similar ubiquity is found with the exponential e to the ax, but that is another story.

Wow that is fascinating and mind boggling how sin waves can do that!
 

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