Can you have sin(L)?

Question

If L is the dimension of length can you have sin(L) or is it just nonsense. If you can have it is the result dimensionless.

Answer 1

Well, all the previous posters are correct in thinking that, in normal life, L cannot be a length.

However, L does not need to be dimensionless. For example, if L expresses the measure of an angle, it will be in radians (or degrees or grades); it will have units. It is the result sin(L) which is dimensionless (no units)


It is also possible in navigation (orthodromy = measuring with great circles on a sphere) to have the length of a triangle's side expressed in units that are 'disguised' angles: a nautical mile is the length of an arc, on Earth's surface, subtended by an angle of 1 minute (1/60 deg.) at Earth's centre -- and the kilometre was also defined with a central angle of 1 centigrade (1/100 of a grade).

So, in spherical geometry, we routinely use trig. functions of 'sides'.

In astronomy, where distance is a touchy measurements (depending on the Hubble constant one decides to use), apparent distances (L = separation between two points in a distant galaxy for example) is given in radians.

The beauty of using radians is that for very small angle, sin(L) = L and you 'avoid' taking the trig. function of a distance (in reality, the trig. function is automatically part of the measurement).


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For periodic phenomena (like radio waves or photons), the phase at which one expects to find something can be linked to the distance.

For example, we have a laser made up of a single wavelength of light, all photons in phase in a coherent beam. If we express the distance down the beam as a fraction of the wavelength (one unit of distance = wavelength divided by 2*pi), then taking the sine of the distance from the beam source to the target may provide something useful. This would work because the distance, expressed in this fashion, automatically gives you the phase-angle of the photons at that exact position. (in other words, you are still taking the sine of an angle -- again, the distance is a disguised angle).

Answer 2

Actually, you CAN have sine or other functions of what appear to be "lengths," BUT NOT IN PLANE GEOMETRY.

In 3D there are many theorems that hold on the UNIT SPHERE. There, the "lengths" of the sides of spherical triangles (those lengths being parts of great circles) are given by the angles they subtend at the centre of the unit sphere. Thus, though appearing visually to be curved lengths analogous to the sides of plane triangles, they are in fact dimensionless.

The theorems you're familiar with in plane geometry all have their analogues in spherical geometry. Indeed, one can regard the plane geometry theorems as SPECIAL LIMITING CASES of the spherical results!

Thus the spherical geometry "Sine Law" theorem:

sin b / sin c = sin B / sin C

naturally leads to:

b / c = sin B / sin C, that is sin B / b = sin C / c.

See if you can think what the form of the spherical theorem might be, that leads to:

a^2 = b^2 + c^2 - 2bc sin A.

[HINT: what are the leading terms of cos x ?!]

These spherical theorems have very practical usage in surveying, and in determining so called "great circle routes" for airplanes and ships.

REMINDER/AMPLIFICATION: Of course, as I implied above, things you're used to seeing in plane geometry as arbitrarily large physical sides of triangles become in effect (dimensionless) small angles a/R etc. for arbitrarily large spheres of radius R, and the "plane limit" of the spherical theorems really comes about by taking the limits a/R etc. --> 0.

Live long and prosper.

Answer 3

In a 30-60-ninety triangle, ratio of factors is a million : ?3 : 2 the place: shorter leg (a million) is opposite attitude 30 longer leg (?3) is opposite attitude 60 hypotenuse (2) is opposite appropriate attitude (ninety) sin(30) = opp/hyp = a million/2 sin(60) = opp/hyp = ?3/2 sinL = a million/2 L = 30 tiers notice: As you learn trigonometry extra, you certainly must be taught sin and cos for here angles: 0, 30, 40 5, 60, ninety. it will make fixing trig equations plenty less complicated.

Answer 4

Can't be done! You can only take the sine of an ANGLE.
Because the sine of an angle A is a RATIO of lengths of sides in a right triangle with one angle equal to A.

Answer 5

Hi,

The input for sine should always be a dimentionless, but you need to remember that sine works for both degrees and radians.

Also, the output for sine will always be dimentionless as it's defined as a ratio.

Hope that helps,
Matt

Answer 6

"L" must be a dimensionless quantity for any of the trig functions to make sense.

Answer 7

No. sin (x) is meaningful only if x is the measure of an angle usually in degrees or radians.