Analitical geometry; angles in Radians, Why ??

I have a better question - why degrees, why grads?

degrees - 360 in a circle

grads - 400 in a circle

radian - 2(pi) in a circle

It makes the math come out nicer. Depending on what your definition of "nicer" is.

Simply because it is convenient for some types of calculation.

The radian is the natural unit for angles and this is the razon for been used at the IS (International System):

If you evaluate the "limit" when x, "tends to zero" x->0 for the function sin(x)/x, where x is an angle expresed in radians the result is ''1''.

If you look at the Taylor series for sin(x) and cos(x) you'll see that those are "valid" only if you use "x" as an angle expresed in radians.

Now, if you're interested in history: the word "radian" was used by first time in 1873. James Thomson at the Queen's College in Belfast. He worte an exam with questions where that word was used. Other people say that Thomas Muir at St. Andrew's University in 1869 used a variation for the word radian like "rad" or "radial".

Best reagrds,

The radian is the natural unit for angles and this is the razon for been used at the IS (International System):

At the risk of being argumentative . I'd say that's tripe...

There is nothing natural about any system of angular measurment...it's all about convenience...

One could argue that 60 degrees should be the basic unit as this is what occurs if you step off around the circumference, with the pair of compasses that have just drawn the circle...

2 x this 60 degree unit gives the angle between each of the three flights on an arrow... and we all know that 3 is a magic number!

Lo and behold the new unit of angular measurement the 'Cat' completely natural, free from all additives and equal to 60 degrees in old money...

I must go and have a nap now ...

prrrrrr

One could argue that 60 degrees should be the basic unit as this is what occurs if you step off around the circumference, with the pair of compasses that have just drawn the circle...

Illistrated:

I agree, when you use radian in elec eng calculus problems the answers come out nicely in terms of pi which often cancel with the pi's used in the area or volume bit. So radian are not absolutely necessary but they tidy some equations up. ffej

I tried to work out what 90 degrees is, in radians, but I got in a right tangle.

pi / 2 or 1.57 something... I don't have a calcliator handy

FFJ has got the right idea - essentially you can crunch the numbers with a garuanteed common denominator. Like everything else, it's arbitrary. Why? Because I said so!

I was involved in the design of a field gun sight a few years back and was puzzled why the 0 to 90 degrees were translated into 0 to 1500 millirads...

The answer I was given is that 1 rad is the elevation of the gun barrel that gives maximum range to target...

So I asked but why are there 1500 millirads to 90 degrees its actually 1571 millirads to 90 degrees?

Are I was told that the number of millirads in 90 degrees was rounded to make it easier for the gunner to add and subtract from to correct his aim...?

So for military use there appears to be a new unit called the radian but its not related to the real radian if you know what I mean....

Puzzled? I still am..... John

All these years I thought 45^o was angle which gives maximum range of a projectile. I think it still is.

I always thought so as well Bill..... maybe he was lying! but he did qualify himself by saying it does depend on the wind direction and force, but close to 60 degrees is the optimum angle to fire a long range field gun such as this, he was talking about 22 miles range...

I didn't question him as he was one of these boffins that knew weapons technology inside out... I spent a whole day chatting with him, a very knowledgeable bloke...

John.

I guess it's hard to shoot 22 miles in a vacuum, so maybe he is right, considering wind and all.

Oooooh you could be starting a good thread with this one....

Some projectiles aren't actually ballistic eg balls . An arrow will fly to some extent... but I can't remember the angle for max range..I'l have to dig out my copy of:-

'Longbow a social and military history' by Robert Hardy ISBN1 85260 412 3 wich has some excellent technical appendices.

Watch this space (not litterally you can go and make a coffee)

Yep. I can't wait to see the differential equations! I wonder now how many will post the correct one(s). Somebody is sure to throw in a table of correction factors or drag coefficients for different projectiles and whether or not Earth's gravity changes significantly from the point of origination to the point of impact, etc., etc.

Yep, I think you may be right that this could get out of hand in a hurry!

On a side note, I wonder which Greek originally spelled Analytical Geometry . . . anal in Greek is spelled the same as anal in English.

Oh yes this guy was talking about the curvature of the earth the fact the projectile is spinning, the gravitational forces and so on...

His whole background was designing naval guns of large calibre. I did ask my dad about this as dad was in the REME in WW11 and worked on field guns etc... He had a whole heap of stories and information about how these field pieces could be made so very accurate over huge distances...

Very interesting stuff....... John.

Bill

I know there've been a few more posts on this, but 45^o is right for "ideal" conditions (no air resistance) as can readily be shown with a quick bit of calculus. But when there is air resistance it seems it's better to fire higher so more of the path is at lower resistance. If range is 22 miles I work out max height (no air resistance case) 5.5 miles, where air density is well down.

Codey

Man did not invent radians, only the name. Man invented gradients and degrees because they seemed convenient. Radians are a "dimensionless" unit which is the ratio of arc length to arc radius (hence there are 2Π radians in a full circle since a full circle arc is the same as the circumference 2Πr)