Triangles come in many shapes. It would be hard to classify triangles of arbitrary shapes, but we note that any triangle ABC can always be divided into two The three angles inside a triangle add up to 180 A + B + 90° = 180° Subtracting 90° from both sides A + B = 90° Given the value of just one angle A, the other angle B is completely determined (it equals 90°-A), and so is the Let the sides of the triangle be named ( a To tell them apart, just remember:
A simple connection exists between the sine and cosine of any angle. For by the theorem of Pythagoras a Therefore, for any angle A (sin A) This statement is usually written with the square written sin sin Both As the angle A gets closer and closer to 90° (and B gets smaller and smaller), the triangle becomes increasingly narrow and skinny. The length of side |

By the way: the first table of sines was compiled by Al-Khorezmi, who lived in Baghdad around 780-850 and who also gave us the term algebra. Today's hand-held calculators display their values at the push of a button.
The origin of the name "sine" (in Latin sinus, a bay) is interesting. Like the decimal system, it originally came from India and was adapted by Arab mathematicians around the time of Al-Khorezmi. They transliterated the Indian name for the sine without its vowels (which Arabs did not write) as . In 1085 the Castillian (Spanish) king Alphonso the 6th captured Toledo from the Arabs, and with it captured a large library with many Arab manuscripts, including translations of Greek books unknown in the rest of Europe. Alphonso hired scholars who gradually translated those books into Latin. jbIn 1145 one of those translators, Robert of Chester, translated Al-Khorezmi's "Algebra." At one point in the book he encountered the word "jb" and, not realizing it was a foreign word transliterated into Arabic, looked up what its Arabic meaning might be. With appropriate vowels added, it meant "bay", which in Latin was "sinus." That was what he wrote down and that is the term still used. It is given its proper meaning in medicine (as in "sinus headache") where it means the cavities ("nasal sinuses") extending from the nose towards the eyes. |

- (The above story comes from page 96 in a small but delightful book "
A History of π (Pi)" by Petr Beckman, St. Martin's Press, 1971.)Other trigonometric functions exist, such as the tangent of A, discussed in section M-12, written /b = sinA/cosA/a = cosA/sinAThe tangent has the |

**Next Stop: #M-9 ****Deriving sines and cosines**

Author and Curator: *Dr. David P. Stern*

Mail to Dr.Stern: **audavstern**("at" symbol)**erols.com** .

Last updated 25 November 2001