Sanderson M. Smith

NOTES RELATING TO BASIC TRIGONOMETRY
(GIVEN TO ALGEBRA II STUDENTS)

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!

ALL SINE AND COSINE VALUES CAN BE FOUND ON THE UNIT CIRCLE.

As a simple example, consider an angle of 311o. 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 311o, sin 311o) = (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.