Getting an angle on angles

The ancient Babylonians counted using a base sixty system. Unfortunately,
this system has survived to today in the numerics we use to count time and
angle. We write time as h:m:s where sixty seconds (s) = one minute (m) and
sixty minutes = one hour (h). Similarly, angles are written as d:m:s with the
same relationships (60 arcseconds = 1 arcminute, 60 arcminutes = 1 degree).
Mathematicians normally add the prefix 'arc' to distinguish the fact that
they're talking about angle and not time. (There really ought to be a special
circle in hell for anyone who uses the same term for two completely disparate
units or, like the American gallon, redefines an existing unit.)

For most practical applications it's much more convenient to express angles as
decimal numbers. This raises the problem of converting between the two
notations.

Going from d:m:s to decimal notation is straightforward. Consider converting
12:34:56 (12 degrees, 34 arcmin, 56 arcsec) to decimal degrees. We know that
34 arcmin is 34/60 of a degree. We also know that there are 60*60 = 3600
arcsec in a degree. So the 56 arcsec is 56/3600 degrees. Adding them, we
have:

12 deg + 34/60 deg + 56/3600 deg = 12.582221... degrees

or, in general form:

d:m:s = d + m/60 + s/3600 decimal degrees

If your print calls out 12:34:56 d:m:s and you need the tangent of that angle
you'll need to perform the above calculation to get the decimal degrees
to feed to the tangent function. (Better scientific calculators have this
conversion built-in but the less expensive ones often lack it.)

Converting from decimal to d:m:s isn't very difficult. Using 12.582221
decimal degrees as an example:

Extract the integer degrees:

12.582221 = 12 + 0.58222112 degrees

Multiply the remainder by 60 (arcmin/deg):

60 * 0.582221 = 34.93326

Extract the integer arcmins:

34.93326 = 34 + 0.9332634 arcminutes

Multiply the remainder by 60 (arcsec/arcmin):

60 * 0.93326 = 55.9956~56 arcseconds

Again, better scientific calculators have a single key to do this conversion.
However, if yours lacks it, no worry. You won't be doing it frequently and
the procedure above is straightforward.

Most scientific calculators can deal with angles in decimal degree notation,
radian notation and grad notation. So, the question arises:

What the hell are radians and why do we need them? Isn't d:m:s notation
confusing enough? Now you're telling me that we need two more ways of
expressing angles?

When doing mathematics, it's much more useful to express angles in a
notation such that the angle so expressed, when multiplied by the radius of a
circle, yields the length of the arc on the circle subtended by that angle.

Consider a 90 deg angle. It subtends one-quarter of the circumference of a
circle or an arc length of 2*pi*r/4 (2*pi*r = the circumference of a circle
whose radius is 'r'). We want this angle (we'll call it 'A') expressed in
radians to satisfy:

A (rad) * r = 2 * pi * r / 4

That is, the angle in this radian notation, multiplied by the radius of the
circle, equals the length of the arc on said circle subtended by this angle.

Cancelling the 'r's, we have:

A (rad) = pi/2 radians

Since we assumed that A=90 deg, we now have a relationship between degrees and
radians.

90 deg = pi/2 radians
or:
1 deg = pi/180 radians =~ 0.017453 rad
or:
1 radian = 180/pi degrees =~ 57.295831 deg

Which makes things pretty simple. If we have degrees and want radians,
multiply degrees by 180/pi. If we have radians and want degrees, multiply
radians by pi/180. Rather than trying to memorize that, simply remember that
a full rotation, 360 deg, equals 2*pi radians.

-------------------------------
For completeness, a brief note about grads.

The French seem never happy with any measurement system they didn't personally
invent. They thought that 90 degrees was an awkward number for a right angle
so they 'metricized' it to be 100 grads. I don't remember the details but
their argument for this aberration revolved around the fact that slopes
expressed in percent (as we express the slope of hills in road-building) would
then convert directly to grads without the need to do any calculation.


Don't worry about grads. In 30+ years of doing mathematics for a living,
I *never* had to convert any angles to grads. Should you ever need to do so,
the relationships are:

100 grads = 90 degrees = pi/2 radians
-------------------------------

Back to radians and the mathematicians. If you ask a mathematician to
calculate the sine of an angle, he'll write down something like this:

sin (x) = x - x^3 / 6 + x^5 / 120 - ...

This is the 'series expansion' for the sine of x and it's only a valid
equation if x is expressed in *radians*. By using enough terms in this series,
you can calculate the sine to whatever precision you desire. In fact, early
calculators used a series similar to this to calculate trig functions.
(Today, with cheaper memory, they use a table lookup scheme.)

Now, if we look at this equation, we can see that, if x is a small number
(i.e., a lot less than one), x^3/6 is a lot less than x and x^5/120 is a lot
less still. In other words:

sin (x) ~= x for x << 1

A numerical example will verify this. Using my calculator:

sin (5 deg) = 0.087155742

5 deg * (pi/180) rad/deg = 0.087266462

In other words, 5 deg expressed in radians is pretty close to the sine of 5
deg. Some other examples:

angle = 1 deg: sine = 0.017452, radians = 0.017453, error = 0.005077 %
angle = 2 deg: sine = 0.034899, radians = 0.034907, error = 0.020311 %
angle = 3 deg: sine = 0.052336, radians = 0.052360, error = 0.045707 %
angle = 4 deg: sine = 0.069756, radians = 0.069813, error = 0.081278 %
angle = 5 deg: sine = 0.087156, radians = 0.087266, error = 0.127037 %
angle = 6 deg: sine = 0.104528, radians = 0.104720, error = 0.183005 %
angle = 7 deg: sine = 0.121869, radians = 0.122173, error = 0.249205 %
angle = 8 deg: sine = 0.139173, radians = 0.139626, error = 0.325666 %
angle = 9 deg: sine = 0.156434, radians = 0.157080, error = 0.412420 %
angle = 10 deg: sine = 0.173648, radians = 0.174533, error = 0.509506 %

-----------------------
Another aside. When physicists write the differential equation for a swinging
pendulum, they assume small angular motions of the pendulum and use this
sin(x) ~= x approximation. This allows them to obtain a simple equation for
the period of the pendulum. The terms of the series they throw away account
for an error that clock builders call the 'circular error' which will cause
timekeeping errors if the pendulum is allowed to swing through too wide an
arc. Ask yourself: Have you ever seen a pendulum clock where the pendulum
swings through a wide arc?

The series expansion for the cosine looks like

cos(x) = 1 - x^2 / 2 + x^4 / 24 - ...

Using the same argument we used for the sine, we see that, if x<<1,

cos (x) ~= 1 for x << 1

Now, given that

tan(x) = sin(x) / cos(x)

we see that, for x<<1, we can use the following approximations...

sin (x) = x
cos (x) = 1
tan (x) = sin (x) = x

where the value of 'x' is expressed in radians.

That the French are never happy is accepted.
That they gave us the first, lasting, and practical vernier is reasonable justification. Along with L. Moinet's chronograph and John Harrison's mechanical adaptation for interpreting longitude...
That the rest of the free world accepts it, is theirs.
That the machinists, mechanics, and engineers make them work together is a marvel, and ours to select.

While the Babylonians instigated a 60 unit base isn't all bad. Ask a navigator.
The 360 degree circle, a sixty-sixty-sixty hour-minute-second of time, the infinitely calculable arc, and 6000 foot nautical mile work pretty damn well together. Often there isn't time for precision underway. The accurate part was utilized in voyage planning (strategic). Avoidance of hazards leaves marginal time for reactive (tactical) measures to be consumed by accuracy. Strategy and tactics as terminology aren't restricted to combat situations.