# trig symbols

This topic is 5408 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

excerpt from my LJ something intresting poped into my head today. why don't thetrig ratios and logrithims and the such have cool symbols, when calculus does? i mean, it's been around longer then calc, i wonder what the egypyians used for those symbols and the such.

##### Share on other sites
I suppose because "sin(x)" is short and simple enough but "integrate(-50, 50, f(x), dx)" is a bit more of a mouthful, and a good candidate for shortening.
EDIT: Especially considering that sin(sin(sin(x))) is pretty rare, whereas ∫∫∫f(x,y,z)dxdydz is not at all uncommon.

##### Share on other sites
Tangent: I read in some book by Richard Feynman (the physicist), that when he "discovered" trigonometry as a child, he made up his own symbols for the trigonometric functions, but he abandoned them since no one else used them.

##### Share on other sites
Quote:
 Original post by MuzzafarathTangent: I read in some book by Richard Feynman (the physicist), that when he "discovered" trigonometry as a child, he made up his own symbols for the trigonometric functions, but he abandoned them since no one else used them.

Cool. I've read several of his books myself, but don't recall this detail. I'll look out for it next time. I know he invented "Feynman Diagrams," which I guess are still in use. But, that's quantum mechanics stuff.

##### Share on other sites
The creation of Trigonometry is interesting in and of itself. If you think about it, Geometry does not measure the angles, etc, but shows relations between them like theta + gamma = something. Someone had to devise a method to apply number to angles and lengths. The trigonometry functions originally were defined to give the lengths of full chords, and not arcs. Even the range of the numbers giving them meaning from 0 to 2PI was arbitrary. People saw that the angles could be given number if the triangle was viewed as inside a unit circle, and thus the angle would go from 0 to 2PIr, since r = 1, the angles are from 0 to 2PI. Even the fact that 0 is measured from the x-axis was arbitray. For computer systems, it is better to have it at the negative x-axis. That is, the branch cut. The system devised works well, but I don't most people really understand that it was created to give number to angles and sides and that it was arbitrary.

Indeed, very interesting!

##### Share on other sites
I might've misunderstood, so forgive me if I have.

I wouldn't say the choice of 2π to represent a whole revolution is entirely arbitrary. There are two reasons I can think of why it's chosen as opposed to any other, like degrees:
1) Representing an angle θ in radians (ie from 0 to 2π) means that the length of the arc made by that angle on a circle of radius r is just l=rθ. This also works for the whole circle, ie the circumference c=θr, with θ=2π. Good ol' 2πr!
2) Using radians is natural in trigonometric calculus. When differentiating or integrating a trig function, if the angles used (assuming they are angles) are not expressed in radians then you'll have to deal with constant factors popping into the equations. Using radians, d(sin x)/dx=cos x, but using degrees for example, d(sin x)/dx=(180 cos x)/π

I would say the choice of the degree system (0 to 360) was arbitrary, but there are some good mathematical reasons to use radians.

[Edited by - Doc on September 26, 2004 6:04:16 PM]

##### Share on other sites
Hi there,

I never said that the choice wasn't a good one. Just that it was arbitrary. It takes a mental construction to relate an angle in a triangle to one for a unit circle. It didn't have to be that way, but it is very useful. The range could have been -1 to 1 units, and be defined using some other criteria. That of a circle comes naturally.

In trigonometry, there are a number of human creations that work together. For instance, if I draw a right-angled triangle and then give numbers to represent the lengths of the sides, I would be applying a creation to that object. That is, I would be giving a number to things that really don't have number. It is the cognitive methaphor that allows me to give a number to a length. It comes to us so naturally that most people don't even recognize it. So, suppose we agree that we can give number to length and I give you that right-angled triangle and the numbers for the lengths of its sides. Suppose the values are 1, 1 and sqrt(2). The sin of an angle first has to be defined, and we agree that is the half the length of the chord of twice the angle. So now we have a basic understanding of what we mean by sin(theta). So we know for our triangle that sin(theta) = 1/SQRT(2). Cosine is defined to be the sin(90 - theta), so cosine(theta) = 1/SQRT(2). Now, that is great, but what is the "measurement" of the angle theta? At least for our right-angle triangle, the range of theta will be between almost nothing and whatever value we assign to the right-angle (for us, 90 degrees or PI/2). I say almost nothing because if theta = 0, then we don't have a triangle. So theta_min < theta < theta_max. What are good values to assign to theta_min and theta_max which will allow us to have theta well-defined and easily measured? Again, this is arbitrary. We just need a good rule/agreement that is practical to use and easily implemented. Let's take the usual approach and visualize our triangle as being within a unit circle. Since a circle's circumference is well-defined, we can agree on how to relate our angle theta to a portion of the circumference. Theta_min and Theta_max will then be well-defined. I'm just treating the first quandrant, the rest in just an extension.

Back to what we know now. We have a right-angled triangle with sides 1, 1, and SQRT(2). Based on geometry we know that two of the angles are equal to eachother. And we know that sin(theta) = 1/SQRT(2). But what is theta? Since we have an agreement on how to assign a number to theta, we do by using geometry and seeing that we are slicing the circle into 8 equal pieces. Since the circumference is 2PI. We agree that theta should then have the value 2PI/8 = PI/4. Therefore, using our creations, we agree that sin(PI/4) = 1/SQRT(2) and cosine(PI/4) = 1/SQRT(2).

So you see, that is what has been created. It is wondeful and practical, has many many uses, and is practical to use. Trigonometry didn't have to be used with sines, it actually originally used full chords. What we call sine is actually half a chord. Sines are better than chords for what we use trigonometry for, and it shows how trigonometry evolved. Even after we define what we mean by sine(theta), we still have to make further agreements on how we will assign a number to theta that is well-defined. Further along, for instance, theta didn't have to be measured starting at the positive x-axis and increase in a counter-clockwise directions. It could have started at we call -45 degrees, or any other location, even after agreeing that the angles will be assigned values based on a relationship to circumference.

Trigonometry is a quite amazing construction! Think of all the mathematics that uses it!

• 21
• 13
• 9
• 17
• 13