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Name: Jerome N.
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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 as:

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 very elegant.

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".

Vince Calder


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 left angle.

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 sin(x/2)cos(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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