Around 140 BCE, the Greek astronomer Hipparchus of Rhodes compiled the first known table of chords — for each angle in a circle, the length of the straight line connecting the two ends of an arc of that angle. He was not doing geometry for its own sake; he was trying to predict the positions of the stars. Trigonometry was born as the servant of astronomy, and for the next two thousand years the two disciplines moved together. The Indian mathematician Aryabhata refined the chord into the modern sine function in the sixth century. The Arabs introduced tangent and cotangent. Regiomontanus in the fifteenth century brought the apparatus to Europe. Leonhard Euler in the eighteenth century crystallized everything into the unit-circle definitions we still teach.
Sine, cosine, and tangent are functions of an angle. On the unit circle — the circle of radius 1 centered at the origin — the angle θ measured counterclockwise from the positive x-axis lands at the point (cos θ, sin θ). Tangent is sin θ / cos θ, the slope of the line from the origin to that point. From this minimum, the rest of trigonometry unspools: the Pythagorean identity sin²θ + cos²θ = 1 (Pythagoras applied on the unit circle); the addition formulas sin(α+β) = sin α cos β + cos α sin β; double-angle and half-angle identities; inverse functions (arcsin, arccos, arctan). The deeper structure becomes visible through Euler's formula: e^(iθ) = cos θ + i·sin θ, the most beautiful equation in mathematics, which says that the trigonometric functions are the real and imaginary parts of a complex exponential. From Euler's formula the bewildering forest of identities becomes a thicket of ordinary algebraic manipulations of exponentials. The connection to Fourier analysis is then immediate: any periodic function can be decomposed into a sum of sines and cosines, which means sines and cosines are the irreducible atoms of periodic phenomena. This is why trigonometry shows up everywhere there are waves: every oscillating system in physics — sound, light, electromagnetic radiation, water, quantum-mechanical wavefunctions — speaks the language of sines and cosines.
Modern audio and image compression (MP3, JPEG, H.265) is built on Fourier transforms, which are built on trigonometry. Computer graphics uses trigonometric functions for every rotation in three-dimensional space. Robotics kinematics — figuring out where the end of a robotic arm is, given the angles of its joints — is applied trigonometry. Music theory and digital audio synthesis describe pitches as frequencies and timbres as combinations of sine waves. GPS, radar, sonar, and seismic imaging all read the world through trigonometric phase relations. The little tables Hipparchus computed by hand twenty-one centuries ago turned out to encode the geometry of nearly every signal humans now process.