Trigonometry for Dopes - What's Your Sine?

Trigonometry is a word which derives from the Greek,own separate language.
meaning "measure of triangles." Its origins can beEven more amazing is that this field of trigonometry
traced back to early Mesopotamia where sailors andcan be learned by understanding some simple
navigators invented this discipline to help determinerelationships, and I guess the most rudimentary ones
position and location. To think an act so simple as theare the sine, cosine, and tangent of a given angle. In
measurement of triangles and the study ofthe simplest of language, the sine is the ratio, or
relationships between its angles and sides couldquotient, formed by the length of the opposite side to
harbinger the development of a mathematical disciplinethe length of the hypotenuse. The hypotenuse is the
which in turn would find itself involved in such diverselongest side of any right triangle, and is the side directly
fields as astronomy, navigation, music theory, acoustics,opposite the right angle-bigger angle opens up to a
optics, electronics, economics, number theory,bigger, or longer side. Thus if we look at a standard
pharmacy, chemistry, geodesy, cartography,right triangle with side lengths of 3, 4, and 5 (which one
crystallography, architecture, oceanography,must be the hypotenuse?), then the sine of the angle
engineering, game theory, ultrasound, cryptography,formed by the hypotenuse and the shorter leg is 4/5
meteorology, seismography, medical imaging (i.e. CATor 0.8 (opposite the shorter leg is the longer leg).
scans), phonetics, statistics, and biology, to name aSimilarly, the sine formed by the hypotenuse and the
few-then you have thought rightly.longer leg is 3/5 or 0.6. That's all there is to it. The
If the list of fields above astounds you, then you areratios for the cosine and tangent-which we shall
not alone. Many people go through life without learningexplore in another article-can be found using the
even the most superficial mysteries of mathematics. IfSOHCAHTOA mnemonic. (See my article "Why
any language is universal, mathematics is the ultimateStudy Math? Trigonometry and SOHCAHTOA")
lingua franca. Mathematics is indeed the language ofOnce we understand these ratios, we can then delve
the universe-the language that is understood andinto the wonderful world of trigonometry and all its
applicable in star systems within galaxies so remoteglorious applications in fields like the ones mentioned in
that the only way to visit them is within the constructsthe first paragraph. Gee, is it any wonder that
of our minds. If this fact does not amaze you then lookmathematics is so fascinating? So remember. The
at a more mundane comparison: in the country of India,next time someone asks you what your sign is, don't
language diversity is so enormous that even differentsay Aries, or Gemini, or Leo; but rather say,
villages only miles away from each other have theirtrigonometry.