Properties of the cosine and sine

The fundamental relationship

For any real number t, the point M(cos(t), sin(t)) lies on the circle of radius 1, so:

cos²(t) + sin²(t) = 1

This relationship allows us, for example, to find sin(t) if we know cos(t) and the sign of sin(t).

Bounds on Values

For any real number t: -1 ≤ cos(t) ≤ 1 and -1 ≤ sin(t) ≤ 1. A calculation result giving cos(t) = 1.5 or sin(t) = -2 is therefore necessarily an error.

Periodicity

cos(t + 2π) = cos(t) and sin(t + 2π) = sin(t) for any real number t: we say that these functions are periodic, with period 2π.

Parity

cos(-t) = cos(t): the cosine is an even function. sin(-t) = -sin(t): the sine is an odd function.

Useful related angles

Relation	cos	sin
π - t	cos(π - t) = -cos(t)	sin(π - t) = sin(t)
π + t	cos(π + t) = -cos(t)	sin(π + t) = -sin(t)
π/2 - t	cos(π/2 - t) = sin(t)	sin(π/2 - t) = cos(t)

A common pitfall

The formula cos(π - t) = -cos(t) is very often confused with cos(π + t) = -cos(t): both do indeed have a minus sign in front of the cosine, but this is not the case for the sine (sin(pi - t) = sin(t) whereas sin(pi + t) = -sin(t)). Always draw a quick diagram of the circle if in doubt.