What's wrong with argument against cos?

In summary, the cosine term is there because fluxes work that way. Without the cosine term, the argument would be invalid.
  • #1
ecklstn36
2
0
I am trying to compute the amount of energy received by the Earth from the sun, by integrating over the Earth's surface. I keep reading that the formula should be:

E = e-max * cos @ * area

e-max is the solar insolation,
@ is the angle between the sun and the perpendicular to the land, @=0 at noon (at equator)

It's the cosine that I am having trouble with. What is wrong about the following argument against a factor of cosine for the received energy?

Cosine changes fastest at cos @ = 0, and slowest at cos @ = 1. That means that at sunrise(SR) and sunset(SS), the amount of sun that the land gets is changing most rapidly, and at noon(N) least rapidly. When I picture a mental image of the Earth rotating about its axis, it seems to me that the change in amount of sun at SR, SS, and N should be the same, should all be at a minimum, and at +/- 45 deg should be at a maximum.

That would correspond to cos^2, not cos.

It also seems that the factor at +/- 45 should be half of the maximum, by symmetry. A factor of cosine means half of e-max occurs at @=60, which seems anti-intuitive.
 
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  • #2
I'm not sure your argument; however, the cosine term there isn't necessarily for rotation. The cosine term is just there because that's how fluxes work. If you want to compute the flux of a vector field through an area, you need to take the dot product of the vector with the area vector (defined to be the vector with length equal to the area and perpendicular to the area).

The definition of dot products means that there is a cosine term (and not a cosine squared term).
 
  • #3
The rotation is irrelevant to the argument. I might as well imagine the Earth to be standing still in this problem.

If you didn't know that calculating fluxes required a cosine term, if you didn't know this was a flux in the first place, how would you argue for a cosine term when trying to calculate the incoming energy for a piece of the Earth's surface? ie if you said

E = e-max * some factor * area

e-max is the solar insolation

how would you argue that "some factor" should be cosine?

Thanks again.
 

What is the argument against the existence of cos?

The main argument against the existence of cos is that it is a purely mathematical concept and has no physical significance. It is argued that cos, along with other trigonometric functions, were created solely for the purpose of solving mathematical problems and have no real-world application.

Why do some people believe cos is not a real thing?

Some people believe that cos is not a real thing because it cannot be observed or measured in the physical world. They argue that everything that exists must be tangible and have a physical manifestation, and since cos does not fit this criteria, it cannot be considered a real thing.

Can cos be proven to exist?

No, cos cannot be proven to exist in the same way that physical objects or phenomena can be proven to exist. It is a mathematical concept that is defined and accepted within the realm of mathematics, but cannot be observed or measured in the same way that physical objects can.

Does the lack of physical existence make cos less important?

Not necessarily. While cos may not have a physical manifestation, it is still a crucial part of mathematical equations and calculations. It allows us to solve complex problems and make accurate predictions, making it an important concept in the field of mathematics.

Is there any evidence to support the existence of cos?

There is no physical evidence to support the existence of cos, as it is a purely abstract mathematical concept. However, its use in solving mathematical problems and its consistency in producing accurate results can be seen as evidence of its validity and usefulness in the field of mathematics.

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