A full circle measures 2π radians, corresponding to the complete circumference of the unit circle.

In nature, numerous phenomena propagate in the form of waves or can be described by the concept of a wave, and these phenomena can be modeled mathematically by tracing their description back to a simple geometric shape: the triangle.

The Babylonians formulated the concept of an angle based on the equilateral triangle, observing the space between two sides as defining one angle.

The Babylonians used a sexagesimal (base-60) number system and applied this to dividing space and time: they divided angles into 60 parts (degrees) and because exactly six angles of 60° form a complete circle, a full circle measures 360°.

When humans later adopted the decimal (base-10) number system based on having 10 fingers, they were dissatisfied with the Babylonian sexagesimal division of space and time, so they changed angle measurement using pi, the ratio of a circle's circumference to its diameter, which is constant regardless of circle size.

Pi is a constant ratio of a circle's circumference to its diameter (C/D) that remains the same regardless of circle size; when rearranged with diameter = 2 × radius, the circumference formula becomes C = 2π × radius, so for a unit circle with radius 1, the circumference is 2π.

The unit circle (a circle with radius 1) was adopted as a template for measuring angles: instead of dividing an angle into discrete parts (like the Babylonian 60 parts), the arc length enclosed by the angle on the unit circle is measured, so a 60° angle corresponds to 1/6 of the circumference, which equals π/3 radians.

A 90° angle corresponds to 1/4 of the unit circle's circumference, which equals π/2 radians; this angle is so frequently used in mathematics that it is marked with a distinctive dot symbol.

The right triangle became the preferred triangle in mathematics (replacing the equilateral triangle) because it can represent all other triangles (any triangle can be constructed from two right triangles), the Pythagorean theorem applies to it, and understanding right triangles enables understanding all triangle types.

In a right triangle, the side opposite the right angle is called the hypotenuse, and the other two sides are called legs; from the perspective of a specific acute angle (alpha), the legs can be further named as the opposite leg (across from alpha) and the adjacent leg (next to alpha).

Six trigonometric ratios can be formed from the three sides of a right triangle (sine, cosine, tangent, cotangent, secant, and cosecant), where the last three are reciprocals of the first three (with swapped numerator and denominator).

Trigonometric ratios (sine, cosine, etc.) are invariant: they produce the same numerical value regardless of the size of the triangle, because changing the ratio requires changing only the numerator or only the denominator of the ratio, which is impossible without also changing the other, unless the angle itself changes.

Because trigonometric ratios change only when the angle alpha changes, they can be understood as functions that depend on alpha; each ratio has been assigned its own function name: sine, cosine, tangent, cotangent, secant, and cosecant.

If the hypotenuse of a right triangle is set to a unit length (length 1), then sine(alpha) simply equals the length of the opposite side, cosine(alpha) equals the length of the adjacent side, and all other trigonometric ratios can be calculated using only sine and cosine.

The unit circle template can be used to measure angle alpha: with a unit hypotenuse and right triangle inscribed in the unit circle, sine(alpha) represents the y-coordinate (opposite side) and cosine(alpha) represents the x-coordinate (adjacent side) of the point on the unit circle.

To visualize trigonometric functions, a coordinate system can be constructed with angle (alpha) on the x-axis and the trigonometric ratio value on the y-axis; as the angle increases from 0 toward 90°, the sine value increases from 0 toward 1.

When tracing the sine function around a complete rotation of the unit circle, the resulting curve resembles a wave; negative values in this representation indicate that the opposite side is in the lower semicircle of the circle.

And that looks like a wave, doesn't it?

The cosine function can be graphed in the same coordinate system as sine, and it has the same wave shape as sine but is offset (phase-shifted) by 90° (or π/2 radians), because the adjacent side is geometrically offset by 90° from the opposite side.

If we understand sine, we also understand cosine because cosine is simply sine shifted by 90°.