The sine and cosine functions, defined on R

A function associated with a triangle

In a right-angled triangle, the sine and cosine are defined only for acute angles. Using the unit circle (a circle of radius 1), we can associate a point M on the circle with every real number x, and thus define two functions:

• the cosine function, denoted cos: x → cos(x)
• the sine function, denoted sin: x → sin(x)

Domain and range

These two functions are defined for every real number x: their domain is the entire set R. However, as M is a point on the unit circle, its coordinates are always between –1 and 1. We therefore have, for any real number x:

–1 ≤ cos(x) ≤ 1 and –1 ≤ sin(x) ≤ 1

Notable values

x	0	π/6	π/4	π/3	π/2	π
cos(x)	1	√3/2	√2/2	1/2	0	-1
sin(x)	0	1/2	√2/2	√3/2	1	0

Fundamental relation

For any real number x: cos(x)² + sin(x)² = 1

Example

If cos(x) = 3/5 and x is in [0; π/2] (so sin(x) ≥ 0): sin(x)² = 1 – (3/5)² = 1 – 9/25 = 16/25, so sin(x) = 4/5

Common pitfall

cos(x) is neither a distance nor an angle: it is a NUMBER, the result of a function, always between –1 and 1. Writing cos(x) = 2 is therefore always incorrect, regardless of the value of x. Similarly, cos(x)² means (cos(x))², never cos(x²).