Sanderson M. Smith

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**TRIGONOMETRY: **

The word has **Greek
**origins. It means "triangle measure,"
coming from *trigonon* (triangle) and *met'ron* (measure).
Interestingly, there is no evidence that the word "trigonometry" was
used by the Greeks, who began to develop mathematics in attempts to
describe the universe as they knew it around 550 B.C. As a contrast,
the word geometry ("earth measurement") was used by the Greeks. You
already know that the six trigonometric functions (sine, cosine,
tangent, cosecant, secant, tangent) are basically ratios. In the
right triangle, the sine of an acute angle is the ratio (length of
opposite side)/hypotenuse. The Greeks saw everything in terms of
whole numbers, and ratios of whole numbers. They definitely used
trigonometric concepts, although there is no historical evidence that
they characterized important ratios by words.

Despite its Greek origins, the word "trigonometry"
did not appear in mathematical literature until 1590. The
**Hindus** and the
**Arabs**, who
pretty much developed our decimal system using 10 digits
(0,1,2,3,4,5,6,7,8,9) also developed trigonometric ideas before the
word "trigonometry" appeared. For instance, the Hindus produced the
word **sine**, which
technically means "chord." If you are curious, you might try to see
the relation between sine and chord of a circle.

[As a side note, many folks think that the number system we now use was developed by the Greeks or Romans. However, we owe our decimal number system to the Hindus and the Arabs. For those who might have thought our number system came from the Romans, I would ask "How would you like to have to multiply and divide with Roman numerals? (Yuk!)]

As mathematicians such as **Copernicus**, **Kepler**, **Galileo** and others attempted to
convince a skeptical audience that we don't live in an earth-centered
universe, trigonometry became important. While this is a tremendous
oversimplification, there is a lot of angular motion in our universe.
We now know our earth is simply a wanderer in space (Oh man, did the
Christian Church initially rebel against this thought!). As
**Newton**,
**Leibniz**,and
others developed the *calculus* and other branches of
mathematics in attempts to explain our very existence, trigonometry
became important. Many branches of modern-day engineering rely
heavily on trig.

OH, THOSE AMAZING FOLKS FROM YORE WHO DEVELOPED THE MATHEMATICAL IDEAS THAT HELP EXPLAIN OUR VERY EXISTENCE IN OUR MATHEMATICALLY-DESIGNED UNIVERSE! LACKING THE COMFORTS AND TECHNOLOGY THAT WE TAKE FOR GRANTED TODAY, THEIR ACCOMPLISHMENTS BOGGLE MY MIND.

OK, I have said in class that if you keep the following simple statement in mind (and understand it), trigonometry in pre-calculus will flow quite easily... if you are willing to think!

As a simple example, consider an angle of
311^{o}. Draw
the angle in standard position, and construct the unit circle with
center at (0,0). Note where the terminal side intersects the
circle. The coordinates of this point of intersection are (cos
311^{o}, sin
311^{o}) =
(0.65606, -0.75471).

"All things which can be known have number, for it is not possible that without number anything can be either conceived or known."

(Philolaus, ca. 425 B.C.)

"The advancement and perfection of mathematics are intimately connected with the prosperity of the State."

(Napoleon Bonaparte, 1769-1821)

"From the intrinsic evidence of His creation, the Great Architect of the universe now begins to appear as a pure mathematician."

(Sir James Hopwood Jeans, 1877-1946)

"This country does not need cynics and skeptics. We need men and women who can dream about things that never were."

(U.S. President John F. Kennedy)

**MATH POWER TO ALL.**

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