64. What's in a Name? The word sine comes from the Latin word sinus, which means "bay" or "cove." It entered the language through a mistake (variously attributed to Gerardo of Cremona or Robert of Chester) in translating the Arabic word "jiba" (chord) as if it were "jaib" (bay). This was due to the fact that the Arabs abbreviated their technical terms, much as we do today. Imagine someone unfamiliar with the technical term "cosecant" trying to reconstruct the English word that is abbreviated by "csc." It might well enter their language as their word for "cascade."
The names for the other trigonometric functions can all be explained.
(a) Cosine means "sine of the complement." Explain why this is a logical name for cosine.
(b) In the figure below, BC is perpendicular to OC, which is a radius of the unit circle. By a familiar geometry theorem, BC is tangent to the circle. OB is part of a secant that intersects the unit circle at A. It lies along the terminal side of an angle of t radians in standard position. Write the coordinates of A as functions of t.
(c) Use similar triangles to find length BC as a trig function of t.
(d) Use similar triangles to find length OB as a trig function of t.
(e) Use the results from parts (a), (c), and (d) to explain where the names "tangent, cotangent, secant," and "cosecant" came from.
