Sine and cosine began as jyā and koṭi-jyā in Gupta-era Indian astronomy
The functions that describe sound waves, oscillating springs and the changing length of the day trace back to the jyā and koṭi-jyā of Indian astronomers in the Gupta period. At heart they are simple ratios: two sides of a right triangle, each compared with its longest side.
Pick an acute angle in a right triangle. Its sine is the side facing the angle divided by the hypotenuse, the side across from the right angle and always the longest; its cosine is the side next to the angle divided by the hypotenuse. Any right triangle containing that angle gives the same answers, because all such triangles are similar, so the ratios belong to the angle itself. In the triangle with two 45-degree angles and legs of 1 unit, the sine and cosine of 45 degrees both equal the square root of 2 divided by 2.
Four relatives follow. Tangent is sine divided by cosine, equivalently the opposite side over the adjacent one, and cotangent is its reciprocal. Cosecant is the reciprocal of sine and secant the reciprocal of cosine, each comparing the hypotenuse with one of the shorter sides.
The functions also solve triangles that lack a right angle. The law of sines says each side divided by the sine of its opposite angle gives the same value, so two angles and one side are enough to recover everything else. The law of cosines finds a missing side from two known sides and the angle between them, and in the special case of a right angle it collapses into the Pythagorean theorem.
To cover every angle, mathematicians turned to the unit circle, a circle of radius one centred on the origin: a ray at a given angle meets it at exactly one point, whose y-coordinate is the sine and x-coordinate the cosine. Vectors offer another route, through the cross and dot products. Modern treatments define both functions as infinite series or as solutions to differential equations, stretching them to any real number and even to complex numbers, and making them natural tools for modelling light waves, yearly temperature swings and sunlight intensity.
Source: Sine and cosine
