Sine and cosine began on triangles, then owned the circle
Trigonometric functions—also called circular or goniometric functions—tie an angle in a right triangle to ratios of two sides. Sine, cosine, and tangent dominate modern use; their reciprocals are cosecant, secant, and cotangent. Unit-circle geometry and power series then stretch the same names across the whole real line and into the complex plane.
Fix an acute angle θ and every right triangle containing it is similar, so each pair of side lengths yields a fixed ratio—six ratios, six functions. Opposite over hypotenuse is sine; adjacent over hypotenuse is cosine; opposite over adjacent is tangent; the others invert those. The two acute angles in a right triangle sum to 90° or π/2 radians, which forces cofunction identities such as cos(90° − θ) equalling sin θ. Abbreviations sin, cos, tan (or tg), sec, csc (or cosec), and cot (or ctg) migrated from prose labels for segments into functional notation like sin(x) as the function concept matured in the seventeenth and eighteenth centuries.
Elementary angle measures often use degrees—90° for a right angle, 360° for a full turn. Analysis prefers radians: on the unit circle an arc of length 1 subtends 1 radian (about 57.3°), and a full turn is 2π radians (about 6.28). Because the radian is dimensionless, one degree equals π/180. With that unit, sin and cos admit definitions via exponentials, power series, or differential equations that never mention triangles, and their derivatives stay simple.
On the unit circle centred at the origin, a ray rotated by θ from the positive x-axis meets the circle at point A = (xA, yA). Cosine is xA and sine is yA, recovering the right-triangle picture when the radius is the hypotenuse and obeying the Pythagorean identity cos²θ + sin²θ = 1. Further intersections with vertical and horizontal lines, and with the tangent at A, supply tangent, cotangent, secant, and cosecant coordinates for any real θ—positive or negative—far past the acute range.
Each of the six functions has an inverse and a hyperbolic analogue. Sciences that lean on geometry—navigation, solid and celestial mechanics, geodesy—deploy them constantly, and Fourier analysis treats them as the simplest periodic building blocks for waves.
Source: Trigonometric functions
