Actually, the inverse function of the tangent function is called
arctangent, or inverse tangent (tan^-1), but the cotangent is a
different function.  Cotangent is run/rise, or opposite over
adjacent sides.  It is the inverse of the value of the tangent
function.

The inverse function is not the mathematical inverse of the value
of the tangent function, it is the inverse operation.  That is,
the tangent function says, "For the given angle, what is the
ratio of the opposite side to the adjacent side?"  The inverse
tangent function asks, "What angle will have this value of its
tangent function?"  In other words, the tangent gives the
numerical value of a ratio, the inverse gives the angle.

I just retired after teaching mathematics (calculus) for the last
twenty years.

    -Raf

Hi,

I drop in and check out this newsgroup from time to time and have been
following this thread.  I'm a Civil Engineer, experienced in road design
(just threw that out to keep the sniping down to a minimum, I'm thin
skinned).

Typically, we design the road inclinations based on what we term grade.
Most municipalities, DOTs, etc., specify maximum grades based upon various
factors such as speed limits, sight distances, local topography, etc.  In my
area (Texas Hill Country) most of the counties specify a maximum grade for
low volume, residential roads of about 10 - 12 percent.  High traffic roads,
or roads expected to carry commercial trucking (highways) usually try to
limit the grade to 5 - 6%.  You also have to look at the benefit to cost
situation.  Sometimes, these grade limits are waived, especially for shorter
distances, due to extenuating circumstances such as topography.  Grades of
over 12% are considered very steep and often cannot be climbed by fire
trucks (tankers), school buses and trash trucks.

Now, to answer your question about what is grade, a number of people had it
correct.  You simply divide the vertical distance by the horizontal, then
multiply by 100.  For instance, taking a 10% grade (very steep) as an
example, you will climb 10 feet over a horizontal distance of 100'  (10/100
x 100 = 10%).  For a 1% slope (relatively flat) your would expect to climb
1' over 100' (1/100 x 100 = 1%).  If you go out and measure your driveway,
and determine that over 50' the drive rises 5', then the driveway's grade
would be 5/50x100=10%.  To give you a visual of typical grades, a
handicapped ramp into a department store is usually 8.33%, a cross slope on
a city sidewalk or subdivision road is about 2%.

Now that said, grades are not the same as degree (of measurement).  A 10%
grade would correspond to a little less than 6 degrees from the horizontal,
and a 1% grade would correspond to a little less than 2 degrees.  When you
talk of slope, it is pretty much the same as grade, but expressed as a
fraction, not a percentage.  So a road with a 10% grade would have  a slope
of 0.10.  But again, in the world of roadway design, we almost always
express roadway inclination as a percent grade.  I believe that in the
Webster definition quoted above, they are using the word "degree" not as an
angular measurement, but as a descriptor of the severity of the inclination.

And if you are not confused enough yet, when we work with grades of less
than 1%, we typically switch to slope, but express it as a ratio.  For
example, most of the slopes on the sides of the roads (where the dirt is)
are designed to a minimum slope.  The steepest slopes typically permitted
for areas that need to be mowed (grass) are no more than 3:1 (3 horizontal
to 1 vertical).  Roadway shoulders are typically no steeper than 6:1.

I hope this helps, sorry it's so long.

Mike

snip

You must travel different roads than I do.  I can't recall every
seeing a road incline expressed in degrees.

Maybe Mike L's comments will clarify.

This topic isn't anything to loose sleep over.

Mickey