Sine and Angle
Name: Jerome N.
It's been a while since I had trig, but a question a metrologist asked me
is "Can I find the measure of an angle without using tables or calculator
if I know the sine of the angle?"
You are actually asking to find the (inverse sin(x) or arcsin(x)). The
problem is this is not a single valued function it repeats from '-1' to and
'+1'. Let's assume that you will restrict 'x' to the range -1 to +1, then
you can use any of a number of infinite series of expansions for 'pi': Such
pi = 4/1 - 4/3 + 4/5 - 4/7 + 4/9 - ...
There are many others as well. Then just by estimating the desired as some
multiple of 'pi' e.g. 0.011*pi and to the multiplication long hand -- not
Isaac Newton derived the direct expression:
arcsin(x) = x + 1/2*(x^3/3) + (1/2)*(3/4)*(x^5/5) + ...
Seek out the paperback book "A History of Pi" by Petr Beckmann for an
entertaining story of this number that continues to fascinate mathematicians
and lay people alike.
See for example, the Sept 1 issue of "Science News".
Assuming the angle is less than 90 degrees, you can find the angle with a
ruler, a protractor, a pencil, and a piece of graph paper. If the angle is
between 0 degrees and 90 degrees, the sine value will be positive. You can
use a right triangle, the basis of trigonometry, to find the angle. Recall,
sin(angle) is the length of the side opposite the angle divided by the
hypotenuse. If we can make a triangle with the correct ratio of
(opposite)/(hypotenuse), we can measure the angle with a protractor.
Choose a good scale for the graph, perhaps 10 centimeters. Draw a
horizontal line across the bottom of the paper(adjacent side). Draw a
vertical line (opposite side) up from your horizontal line, near the right
edge of the paper of length equal to (sine value)*(scale value). If you
choose 10cm as your scale, this line will be (sine value)*(10cm). Draw a
diagonal line (hypotenuse) from the top of your vertical line down to your
horizontal line. Rotate your ruler before drawing to make sure your
hypotenuse will be exactly as long as your scale value (for me, 10cm). Now,
your opposite side and hypotenuse are correct lengths. Measure the lower
Dr. Ken Mellendorf
Illinois Central College
Sure, you can, by using some approximation.
For example, sin(x)= x - x^2/2! + x^3/3! - ...
This series converges fast is x (in radian) is small. If it is large, you can
first determine sine and cosine of x/2 and then use sin(x)=2
In other words, if you remember some of the formulae, you can obtain the
values approximately without a calculator or a table.
Ali Khounsary, Ph.D.
Advanced Photon Source
Argonne National Laboratory
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Update: June 2012