Trigonometric Functions

Trigonometric functions describe periodic relationships between angles and side ratios. They appear everywhere—from waves and oscillations to signal processing, navigation, and geometry.

The chapter works through three things. It starts with why calculus formulas come out clean only in radians, and how to convert between the two scales. It then takes the six functions one at a time—sine, cosine, tangent, cotangent, secant, and cosecant—giving the definition, domain and range, graph, and periodicity of each. The third piece is the power-reduction identities, which rewrite powers of sine and cosine in a form that is useful when integrating or simplifying.

The names carry a history worth knowing, since it explains them. The ancestors of sine are the chord tables the Babylonians used for astronomy and measurement around 1800 BC; the true prototype is the half-chord table (“jya”) that Aryabhata set down in the fifth century; Arabic scholars such as Al-Battani carried this tradition forward and completed the tables for sine, cosine, and tangent; “jya” travelled into Arabic as “jiba”, was misheard as “jaib” (bay), and became “sinus” in Latin and “sine” in English. Regiomontanus and Viète systematised trigonometry in the fifteenth and sixteenth centuries, which is where cosine (“complementi sinus”, the sine of the complement) and tangent (“tangens”, touching) come from; Euler fixed sin, cos, tan, cot, sec, and csc as the standard six symbols in the eighteenth century and extended the functions to the complex plane.

If you have not yet seen why radians make the calculus formulas clean, start with the section on angles and radians; everything after it assumes that convention.