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Frank wrote:
>> As an aside,  here's a little mantra: Angles *ARE* Ratios.
>> An angle of one arcsecond  is a ratio of 1:206,265. An angle of one
>> arcminute is a ratio of 1:3438. And an  angle of one degree is a ratio of
>> 1:57.3. >> Memorize any one of these and you  never need trig for small angle
>> calculations.

Bill wrote
> Thanks.  If I recall, arc divided by radius yields sine, tangent and angle (in
> radians) in small angles. Your factor of 1:57.3 simply converts from rads to
> degrees. Yes? (Thank you, Bill)

Peter asked Frank:

> Speaking of matters brought up recently, and since Frank seems to be online,
> any chance of some more detail on this issue?

Peter

Not Frank. Bit I'll take a whack at it.  I think it is a two-part question.

1.  Say we purchase 1/8 scale model of a boat.  A 1:8 ratio.  We expect 1
unit on our model to equal 8 units on the real boat.  If the real boat is 24
ft long, our model should be 3 feet long. The 1:8 ratio may also be
expressed as a fraction (1/8) or as 0.125.

A degree is a fraction of a circle.  1/360.
A radian is a fraction of a circle 1/6.28319

Angles can be expressed as ratios:
Sine = opposite side / radius
Tangent = opposite side / adjacent side
Cosine = adjacent side / radius
Angle in radians = length of arc / radius
Those ratios can be expressed as a single number of course.

2.  Small angles (less than 10 degrees)

Let's assume a circle with a radius of 1 unit.
Its circumference will be pi diameter or pi 2r = 6.28319 units

Let's use the above 3-degree angle as an example. A 3-degree angle is 3/360
or 1/120 of a circle.  (It is also 0.052360 rads.)

The length of the arc for that angle is (3/360)*6.283 = 0.052360
Therefore the angle in radians is 0.052350 / 1 = 0.052350

Now let's look at the sine of 3 degrees (opposite / hypotenuse) and solve
for the opposite side:  sin 3 * 1 = opposite = 0.052336

Now that we have the opposite side lets solve for the adjacent leg. Tangent
of 3 degrees = 0.052408. Adjacent = 0.052336 / tan3 = .998630

Now lets compare the angle in radians, sine and tangent of 3 degrees.
0.052350 radians
0.052336 sine
0.052408 tangent

Let's also compare the length of the arc to the opposite side. The length of
the arc is approximately equal to the opposite side, as is the adjacent leg
to the radius. So for a small angle, the sine, tangent and angle in radians
are approximately equal.  The difference between using the small angle
shortcut and doing rigorous calculations is only a fraction of a percent.

Recalling there are = 6.28319 radians in a circle and 360 degrees, we know
that the conversion factor from radians to degrees is 360 / 6.28319 =
57.2958.  The other two constants Frank mentioned would be conversion of
radians to minutes or seconds of an arc.

Hope that helps

Bill

   

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