Below are some abstracts from Trigonometry

Question: An American student is writing by e-mail to her friend in France, and they are doing homework together. The American student writes to the French student: "Look at page xxx [this page] of the Gelfand-Saul trigonometry book. Let's get the sine of angle D." The French student measured EF with his ruler, then measured ED, then took the ratio EF/ED and sent the answer to his American friend. A few days later, he woke up in the middle of the night and realized, "Sacre bleu! I forgot that Americans use inches to measure lengths, while we use centimeters. I will have to tell my friend that I gave her the wrong answer!" What must the French student do to correct the answer?

Solution: He does not have to do anything: the answer is correct. The sine of an angle is a ratio of two lengths, which does not depend on any unit of measurement. For example, if one segment is double another when measured in centimeters, it is also double the other when measured in inches.

Ptolemy's Theorem: Quadrilateral ABCD can be inscribed in a circle if and only if AB*CD + AD*BC = AC*BD; that is, if and only if the sum of the products of the opposite sides equals the product of the diagonals.

In the diagram:

a + b + c + d = (3.14...)
AB = sin c
BC = sin a
CD = sin b
AD = sin d
AC = sin (a + c) = sin (b + d)
BD = sin (a + b) = sin (c + d)
The diameter of the circle equals 1

For any four angles a, b, c, d, such that a + b + c + d = (3.14...), we can construct a similar diagram.
Therefore, Ptolemy's theorem tells us that:

(sin c * sin b) + (sin d * sin a) = sin (a + c) * sin (a + b)

for any four angles whose sum is pi (3.14...). This is a trigonometric form of Ptolemy's Theorem.

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