TriG

Trig, short for trigonometry, is a branch of mathematics that studies the relationships between the angles and sides of triangles and extends to periodic phenomena through the unit circle. It originated in ancient astronomy and geometry, with substantial development by the Greeks and later refinement through the use of radians, a natural unit for calculus.

The core functions are sine, cosine, tangent, and their reciprocals cosecant, secant, and cotangent. For a right triangle, sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. On the unit circle, sine and cosine are the y and x coordinates of a point, and tangent is the ratio y/x, defined when cos is not zero. Angles may be measured in radians or degrees; radians are standard in calculus because the functions have simple periodic behavior with period 2π.

Key identities include sin^2 x + cos^2 x = 1, tan x = sin x / cos x, and the reciprocal relations csc x = 1/sin x, sec x = 1/cos x, cot x = cos x / sin x. Additional results cover angle addition and subtraction, double-angle, and product-to-sum formulas. Inverse trigonometric functions arcsin, arccos, and arctan recover angles from sine, cosine, or tangent values, with principal value ranges chosen to yield unique answers.

Applications span solving triangles and modeling periodic phenomena such as waves, with use across physics, engineering, computer graphics, navigation, and astronomy. Trigonometric functions also underpin rotations, Fourier analysis, and many geometric and analytic techniques, providing a foundational toolset in mathematics and applied sciences.